Instanton Floer homology, sutures, and Euler characteristics

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $\chi_{\rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $\chi_{\rm gr}(SHI(M,\gamma))=\chi_{\rm gr}(SFH(M,\gamma))$ for any balanced sutured manifold $(M,\gamma)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $\chi_{\rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $\widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $\mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^\sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $\underline{\rm KHI}^-(Y,K)$ introduced by the first author.


Introduction
Sutured manifold theory was introduced by Gabai [Gab83] and Floer theory was introduced by Floer [Flo88]. They are both powerful tools in the study of 3-manifolds and knots. The first combination of these theories, called sutured Floer homology, was introduced by Juhász [Juh06] based on Heegaard Floer theory, with some pioneering work done by Ghiggini [Ghi08] and Ni [Ni07]. Later, Kronheimer and Mrowka made parallel constructions in monopole (Seiberg-Witten) theory and instanton (Donaldson-Floer) theory [KM10b]. Different versions of Floer theories have different merits. For example, Heegaard Floer theory is more computable, while instanton theory is closely related to representation varienties of fundamental groups. Hence it is important to understand the relationship between different versions of Floer theories. In this line, Lekili [Lek13], Baldwin and Sivek [BS20b]) proved that sutured (Heegaard) Floer homology is isomorphic to sutured monopole homology, though the relation to sutured instanton homology is still open. In particular, for a knot K in a closed oriented 3-manifold Y , there are isomorphisms I 7 pY qy HF pY q b C and KHIpY, Kq -{ HF KpY, Kq b C.
In this paper, instead of studying the full homologies, we study their graded Euler characteristics, and obtain the following theorem.
Then " means the equality holds for elements in ZrHs{˘H.
The graded Euler characteristic χ gr pSF HpM, γqq was studied by Friedl, Juhász, and Rasmussen [FJR09]. Applying their results, we can relate the graded Euler characteristics of links with classical invariants obtained from fundamental groups.
Consider a finitely generated group π " xx 1 , . . . , x n |r 1 , . . . , r k y. Let H " H 1 pπq{Tors be the abelianization of π modulo torsions. For a generator x i and a word w, let Bw{Bx i be the Fox derivative of w with respect to x i . Equivalently, it satisfies the following conditions.
(1) For any word w " u¨v, we have Bw Bxi " 1 and Bxj Bxi " 0 for any j ‰ i. Consider A " tBr j {Bx i u i,j as a matrix with entries in ZrHs by the projection map Zrπs Ñ ZrHs. Let Epπq be the ideal generated by the minor determinant of A of order pn´1q. Since ZrHs is a unique facterization domain, one can consider the greatest common divisor of the elements of Epπq, which is well-defined up to multiplication by a unit in˘H. This is denoted by ∆pπq and called the Alexander polynomial of π (c.f. [Tur02]). For a 3-manifold M , the Alexander polynomial of M is defined by ∆pM q :" ∆pπ 1 pM qq. For an n-component link L in S 3 , we write t 1 , . . . , t n for homology classes of meridians of components of L and define ∆ L pt 1 , . . . , t n q :" ∆pS 3 zintN pLqq as the multi-variable Alexander polynomial of L. If n " 1 and L " K is a knot, we can fix the ambiguity of˘H by assuming ∆ K pt 1 q " ∆ K pt´1 1 q and ∆ K p1q " 1. In this case, we call it the symmetrized Alexander polynomial of K.
Theorem 1.4. Suppose M is a compact manifold whose boundary consists of tori T 1 , . . . , T n with b 1 pM q ě 2. Suppose where " means the equality holds for elements in ZrHs{˘H.
Remark 1.5. A similar result to Theorem 1.4 has been proved for link Floer homology in Heegaard Floer theory by Ozsváth and Szabó [OS08]. For instanton theory, the case of single-variable Alexander polynomial for links in S 3 were understood by Kronheimer and Mrowka [KM10a] and independently by Lim [Lim09], while the case of the multi-variable polynomial was unknown before.
For knots, the corresponding corollary is the following.
Theorem 1.6. Suppose K is a knot in a closed oriented 3-manifold Y . Suppose Y pKq :" Y zintN pKq is the knot complement and b 1 pY pKqq " 1. Let rms P H " H 1 pY pKq; Zq{Tors -Zxty be the homology class of the meridian of K. Define KHIpY, Kq similarly to KHIpLq as in (1.3). Then we have χ gr pKHIpY, Kqq " ∆pY pKqq¨r ms´1 t´1 P ZrHs{˘H.
An application of Theorem 1.6 is to compute the instanton knot homology of some special families of knots. In [LY20], the authors proved the following. Obviously, a lower bound of dim C KHIpY, Kq can be obtained from χ gr pKHIpY, Kqq. If this lower bound coincides with the upper bound from Theorem 1.8, then we figure out the precise dimension of KHIpY, Kq. This trick applies to p1, 1q-knots in S 3 which are either homologically thin knots or Heegaard Floer L-space knots. In [LY20] we worked with knots in S 3 because prior to the current paper, the graded Euler characteristic of instanton knot homology were only understood in that case. On the other hand, in [Ye21], the second author discovered some families of p1, 1q-knots in general lens spaces whose dim F2 { HF KpY, Kq is determined by χp { HF KpY, Kqq. Hence, with Theorem 1.6 and results in [Ye21], we conclude the following. Corollary 1.9. Suppose Y is a lens space, and K Ă Y is a p1, 1q-knot such that the followings hold. Remark 1.11. The condition H 1 pY pKq; Zq -Z is necessary since terms related to Euler characteristics of torsion spin c structures may cancel out when we consider the map between group rings induced by the projection H 1 pY pKq; Zq Ñ H 1 pY pKq; Zq{Tors. Now, we explain the rough idea to prove Theorem 1.2. First let us consider the case of a closed oriented 3-manifold Y . The Euler characteristic of the framed instanton Floer homology, χpI 7 pY qq, was understood by Scaduto [Sca15,Section 10]. The strategy is to carry out an induction on the cardinality of H 1 pY ; Zq using exact triangles. The grading behavior of χpI 7 pY qq under a surgery exact triangle was fully described as in [KM07,Section 42.3] and it is known that χpI 7 pY 1 qq " 1 forthe any integral homology sphere Y 1 . Hence we can prove χpI 7 pY qq " |H 1 pY ; Zq| inductively. However, the above argument fails when we take into account gradings associated to surfaces inside 3-manifolds. Suppose R Ă Y is a closed homologically essential surface. Then R induces a Z-grading on I 7 pY q by considering the generalized eigenspaces of the linear action µpRq on I 7 pY q (c.f. [KM10b,Section 7]). When trying to understand the graded Euler characteristic in this case, the previous strategy does not apply directly. The reason is that, the surgery curves inducing the exact triangles may have nontrivial algebraic intersections with the surface R, so the maps in surgery exact triangles may not preserve the grading associated to R. We are faced with the same problem when proving Theorem 1.2.
Our strategy is the following. Suppose pM, γq is a balanced sutured manifold and suppose S 1 ,. . . , S n are properly embedded surfaces in M . Then S 1 , ..., S n induce a Z n -grading on SHIpM, γq. After attaching product 1-handles along BM , we can find a framed link in the interior of the resulting manifold such that the link is disjoint from all the surfaces. Moreover, surgeries along the link with all slopes chosen in t0, 1u produce only handlebodies. Since the surgery link is disjoint from the surfaces S 1 , . . . , S n that induce the Z n -grading, the maps in surgery exact triangles preserve the grading. Hence, it suffices to understand the case of sutured handlebodies. In this case, we can further use bypass exact triangles to reduce any sutured handlebodies to product sutured manifolds. It is known that the Floer homology of any product sutured manifold is one-dimensional. Since the behavior of Euler characteristics under bypass exact triangles and surgery exact triangles are the same for both instanton theory and Heegaard Floer theory, we finally conclude that these two versions of Floer theories must have the same graded Euler characteristic.
In the above argument, it is not necessary to treat instanton theory and Heegaard Floer theory separately. Instead, we only use some formal properties that are shared by both theories and hence we can deal with them at the same time. This observation can be made more general. In Kronheimer and Mrowka's definition of sutured (monopole or instanton) Floer homology, they constructed a closed 3-manifold, called a closure, out of a balanced sutured manifold in a topological manner, and defined the Floer homology for a balanced sutured manifold to be some direct summands of the Floer homology of its closure. Then they used the formal properties of monopole theory and instanton theory to show that the construction is independent of the choice of the closures. In the following series of work [BS15, BS16a, BS16b, Li19, GL19], most arguments were also carried out based on topological constructions and hence only depend on the formal properties of Floer theories.
In this paper, we summarize the necessary properties of Floer theory that are used to build a sutured homology for balanced sutured manifolds. In Subsection 2.1, we state three axioms for p3`1q-TQFTs (functors from cobordism categories to categories of vector spaces) called ‚ (A1) the adjunction inequality axiom; ‚ (A2) the surgery exact triangle axiom; ‚ (A3) the canonical Z 2 (mod 2) grading axiom.
A p3`1q-TQFT satisfying these axioms is called a Floer-type theory. For any Floer-type theory H and any balanced sutured manifold pM, γq, we construct a vector space SHpM, γq, called the formal sutured homology of pM, γq, over a suitable coefficient ring. More precisely, we have the following theorem.
Theorem 1.12. Suppose H is a p3`1q-TQFT and pM, γq. If H satisfies Axioms (A1) and (A2), then there is a vector space SHpM, γq associated to any balanced sutured manifold pM, γq, welldefined up to multiplication by a unit in the base field F. Suppose S 1 , . . . , S n are properly embedded admissible surfaces inside pM, γq. Then there exists a Z n -grading on SHpM, γq induced by these surfaces. Equivalently, we have SHpM, γ, pS 1 , . . . , S n q, pi 1 , . . . , i n qq.
Furthermore, if H satisfies Axiom (A3), then there exists a relative Z 2 -grading SHpM, γq, respecting the decomposition in (1.4). Moreover, the graded Euler characteristic χ gr pSHpM, γqq, defined similarly to (1.1) and determined up to multiplication by a unit in˘H 1 pM q{Tors, is independent of the choice of the Floer-type theory.
Remark 1.13. A priori, the definition fo formal sutured homology depends on a large and fixed integer g, which is the genus of the closure; see the Convention after Definition 2.17.
Remark 1.14. The construction of SH is essentially due to work of Kronheimer and Mrowka [KM10b]. Note that instanton theory, monopole theory and Heegaard Floer theory all satisfy Axioms (A1), (A2), and (A3) with coefficients C, F 2 and F 2 , respectively, up to mild modifications (c.f. Subsection 2.1). Moreover, Axioms (A1), (A2), and (A3) are not limited by the scope of gauge-theoretic theories mentioned above and may hold for other more general p3`1q-TQFTs.
There is one further step to prove Theorem 1.2 from Theorem 1.12. For Heegaard Floer theory, the construction coming from Theorem 1.12 is different from the original version of sutured (Heegaard) Floer homology defined by Juhász [Juh06]. It has been shown by Lekili [Lek13], Baldwin and Sivek [BS20b] that these two constructions coincide with each other. Although not shown explicitly, their proofs also imply that the isomorphism between these two constructions respects gradings. Based on their work, we show the following proposition.
Proposition 1.15. Suppose pM, γq is a balanced sutured manifold and suppose H " H 1 pM q{Tors. Suppose SHF is the sutured homology for balanced sutured manifolds constructed in Theorem 1.12 for Heegaard Floer theory. Then we have χ gr pSHFpM, γqq " χ gr pSF HpM, γqq P ZrHs{˘H.
Other than Theorem 1.12, there are more results that can be derived from axioms and formal properties of the formal sutured homology SH. Since the proofs in [BLY20,LY20] are only based on those formal properties, all results can be applied to SH without essential changes. In particular, the following theorem is just the main theorem of [BLY20], replacing SHI by SH. Combining the lower bound from Theorem 1.12 and the upper bound from Theorem 1.16, we obtain the following corollary.
Next, we discuss on the Z 2 -grading on SHpM, γq. Following [KM10b], to construct SHpM, γq, we first construct a closure Y from pM, γq. From a fixed balanced sutured manifold pM, γq, we can construct infinitely many different closures (with the same genus), and the Floer homology of each closure has its own (absolute) Z 2 -grading. Although we can construct isomorphisms between the Floer homology of different closures, the maps do not necessarily respect the Z 2 -gradings. See [KM10a, Section 2.6] for a concrete example. Thus, we cannot obtain a canonical Z 2 -grading on SHpM, γq and the Euler characteristic can only be defined up to a sign (since we do know the maps between closures are homogenous with respect to the Z 2 -gradings).
However, if we focus on balanced sutured manifolds whose underlying 3-manifolds are knot complements, it is possible to obtain a canonical Z 2 -grading. The idea is to compare closures of a general knot with closures of the unknot in S 3 and then fix the relative Z 2 -grading. As an application, for a knot K in a closed oriented 3-manifold Y , we obtain a canonical Z 2 -grading on KHIpY, Kq. Moreover, this canonical Z 2 -grading also carries over to the following decomposition of I 7 pY q introduced by the authors.
Theorem 1.18 ([LY20, Theorem 1.10]). Suppose Y is a closed 3-manifold, and K Ă Y is a nullhomologous knot. Suppose p Y is obtained from Y by performing the q{p surgery along K with q ą 0. Then there is a decomposition up to isomorphism associated to the knot K and the slope q{p.
Proposition 1.19. Under the hypothesis and the statement of Theorem 1.18, there is a well-defined Z 2 -grading on I 7 p p Y , iq. Moreover, for i " 0, . . . , q´1, we have Corollary 1.20. Suppose K is a knot in an integral homology sphere Y . Suppose further that r " q{p is a rational number with q ą 0. Then, the 3-manifold Y r pKq is an instanton L-space (i.e., dim C I 7 pY q " |H 1 pY ; Zq|) if and only if for i " 0, . . . , q´1, we have Proof. If for i " 0, . . . , q´1, we have I 7 pY r pKq, iqq -C, then it follows directly from Theorem 1.18 that Y r pKq is an instanton L-space. Now suppose Y r pKq is an instanton L-space. Applying Proposition 1.19 to Y , we have (1.5) dim C I 7 pY r pKqq ě |χpI 7 pY r pKqq| " | q´1 ÿ i"0 χpI 7 pY r pKq, iqq| " | q´1 ÿ i"0 χpI 7 pY qq| " q.
Hence the inequality in (1.5) is sharp, which implies dim C I 7 pY r pKq, iq " 1.
The techniques to prove Proposition 1.19 can also be applied to study the minus version of instanton knot homology KHI´, which was introduced by the first author in [Li19, Section 5].
Proposition 1.21. Suppose K Ă S 3 is a knot and ∆ K ptq is the symmetrized Alexander polynomial of K. Then there is a canonical Z 2 -grading on KHI´p´S 3 , Kq. Furthermore, we have Conventions. If it is not mentioned, homology groups and cohomology groups are with Z coefficients, i.e., we write H˚pY q for H˚pY ; Zq. A general field is denoted by F and the field with two elements is denoted by F 2 .
If it is not mentioned, all manifolds are smooth and oriented. Moreover, all manifolds are connected unless we indicate disconnected manifolds are also considered. This usually happens When we discussing cobordism maps from a p3`1q-TQFT.
Suppose M is an oriented manifold. Let´M denote the same manifold with the reverse orientation, called the mirror manifold of M . If it is not mentioned, we do not consider orientations of knots. Suppose K is a knot in a 3-manifold M . Then p´M, Kq is the mirror knot in the mirror manifold.
For a manifold M , let intpM q denote its interior. For a submanifold A in a manifold Y , let N pAq denote the tubular neighborhood. The knot complement of K in Y is denoted by Y pKq " Y zintpN pKqq.
For a simple closed curve on a surface, we do not distinguish between its homology class and itself. The algebraic intersection number of two curves α and β on a surface is denoted by α¨β, while the number of intersection points between α and β is denoted by |α X β|. A basis pm, lq of H 1 pT 2 ; Zq satisfies m¨l "´1. The surgery means the Dehn surgery and the slope q{p in the basis pm, lq corresponds to the curve qm`pl.
A knot K Ă Y is called null-homologous if it represents the trivial homology class in H 1 pY ; Zq, while it is called rationally null-homologous if it represents the trivial homology class in H 1 pY ; Qq. We write Z n for Z{nZ.
An argument holds for large enough or sufficiently large n if there exists a fixed N P Z so that the argument holds for any integer n ą N .
Organization. The paper is organized as follows. In Section 2, we introduce three axioms to define formal sutured homology for balanced sutured manifolds and prove the first part of Theorem 1.12. Moreover, we state many useful properties for the proof of the second part of Theorem 1.12. In Section 3, we discuss the modification of Heegaard Floer theory to make it suitable to formal sutured homology and prove Proposition 1.15. In Section 4, we prove the second part of Theorem 1.12. In Section 5, we construct a canonical Z 2 -grading for balanced sutured manifolds obtained from knots in closed 3-manifolds and prove Proposition 1.19 and Proposition 1.21.

Axioms and formal properties for sutured homology
In this section, we construct formal sutured homology and prove some basic properties.
2.1. Axioms of a Floer-type theory for closed 3-manifolds.
Let Cob 3`1 be the cobordism category whose objects are closed oriented (possibly disconnected) 3-manifolds, and whose morphisms are oriented (possibly disconnected) 4-dimensional cobordisms between closed oriented 3-manifolds. Let Vect F be the category of F-vector spaces, where F is a suitably chosen coefficient field. A p3`1q dimensional topological quantum field theory, or in short p3`1q-TQFT, is a symmetric monoidal functor For a closed oriented 3-manifold Y , we write HpY q for the related vector space, called the H-homology of Y . For an oriented cobordism W , we write HpW q for the induced map between H-homologies of boundaries, called the H-cobordism map associated to W . If H is fixed, then we simply write homology and cobordism map for H-homology and H-cobordism map, respectively. Note that by the definition of the involved categories, we have HpY 1 \ Y 2 q " HpY 1 q b F HpY 2 q and Hp´Y q -Hom F pHpY q, Fq.
It is well-known that Floer theories are special cases of p3`1q-TQFTs. Summarized from known Floer theories, we propose the following definition.
A2. Surgery exact triangle. Suppose M is a connected compact oriented 3-manifold with toroidal boundary. Let γ 1 , γ 2 , γ 3 be three connected oriented simple closed curves on BM such that Let Y 1 , Y 2 , and Y 3 be the Dehn fillings of M along curves γ 1 , γ 2 , and γ 3 , respectively. Then there is an exact triangle (2.1) Moreover, maps in the above triangle are induced by the natural cobordisms associated to different Dehn fillings.
Remark 2.2. It is worth mentioning that Axioms (A1) and (A2) are enough for defining formal sutured homology for balanced sutured manifolds. The following Axiom (A3) is only involved when considering Euler characteristics.
A3 Z 2 -grading. For any closed oriented 3-manifold Y , there is a canonical Z 2 -grading on HpY q, denoted by HpY q " H 0 pY q ' H 1 pY q.
A3-3. Suppose W is a cobordism from Y 1 to Y 2 . Then HpW q is homogeneous with respect to the canonical Z 2 -grading. Its degree can be calculated by the following degree formula (2.2) degpHpW qq " 1 2 pχpW q`σpW q`b 1 pY 2 q´b 1 pY 1 q`b 0 pY 2 q´b 0 pY 1 qq pmod 2q. is not essential. Assuming HpY, rΣ g s, 2g´2q " H 0 pY, rΣ g s, 2g´2q shifts the canonical Z 2 -grading for all 3-manifolds. It is worth mentioning that two existing discussions, [LPCS20] and [KM11b], on this Z 2 -grading adopted different normalizations.
The degrees of the maps in Axiom (A2) were described explicitly by Kronheimer and Mrowka [KM07, Section 42.3]. For the convenience of later usage, we present the discussion here. In the surgery exact triangle (2.1), we can determine the parities of the maps f 1 , f 2 , and f 3 as follows.
(1) If there is an i " 1, 2, 3 so that γ i¨δ " 0, then f i´1 is odd and the other two are even.
(2) If γ i¨δ ‰ 0 for any i " 1, 2, 3, then there is a unique j P t1, 2, 3u so that γ j¨δ and γ j`1¨δ are of the same sign. Then the map f j is odd and the other two are even.
Here the indices are taken mod 3.
With Proposition 2.4, the following lemma is straightforward.
Lemma 2.5. In the surgery exact triangle (2.1), after arbitrary shifts on the canonical Z 2 -gradings on HpY i q for all i " 1, 2, 3, exactly one of the following two cases happens.
(2) If there exists i " 1, 2, 3 so that f i is odd and the other two are even, then χpHpY i´1 qq " χpHpY i qq`χpHpY i`1 qq.
Here the indices are taken mod 3.
Remark 2.6. If there are no shifts, then case (2) in Lemma 2.5 happens due to Proposition 2.4.
In this paper we discuss three Floer theories, namely, instanton (Donaldson-Floer) theory, monopole (Seiberg-Witten) theory, and Heegaard Floer theory. However, for any of these theories, we need some modifications as follows. Suppose Y is an object of Cob 3`1 and W is a morphism of Cob 3`1 .
Instanton theory. We consider the decorated cobordism category Cob 3`1 ω rather than Cob 3`1 . The objects are admissible pairs pY, ωq, where ω Ă Y is a 1-cycle such that any component of Y contains at least one component of ω. The admissible condition means that for any component Y 0 of Y , there exists a closed oriented surface Σ Ă Y 0 such that gpΣq ě 1 and the algebraic intersection number ω¨Σ is odd. Morphisms are pairs pW, νq, where ν is a 2-cycle restricting to the given 1-cycles on BW . The H-homology and the H-cobordism map are denoted by I ω pY q and IpW, νq (c.f. [Flo90]), respectively.
The coefficient field is F " C. The decorations ω and ν do not influence Axiom (A1), where the Z-grading is induced by the generalized eigenspaces of pµpαq, µpptqq actions for α P H 2 pY q (c.f. [KM10b,Section 7]).
In the original statement of the surgery exact triangle in [Flo90], different 3-manifolds in the surgery exact triangle may have different choices of ω. However, by the argument in [BS20a, Section 2.2], we can assume that, in Axiom (A2), the 1-cycle ω is unchanged in all manifolds involved in the triangle.
The canonical Z 2 -grading for instanton theory was discussed by Kronheimer and Mrowka [KM10a, Section 2.6]. Indeed, the degree formula (2.2) is from their discussion.
Monopole theory. The H-homology and the H-cobordism map areHM ‚ pY q andHM ‚ pW q (c.f. [KM07]), respectively. Although we useHM ‚ for monopole theory, p3`1q-TQFTs associated to other versions of monopole Floer homology z HM ‚ and HM ‚ are also implicitly used in the proof of the Floer's excision theorem (c.f. [KM10b, Section 3]), which is important to show the sutured homology for balanced sutured manifolds is well-defined. The Z-grading in Axiom (A1) is induced by xc 1 psq, αy for s P Spin c pY q and α P H 2 pY q (c.f. [KM10b, Section 2.4]).
The coefficient field is F " F 2 . This is because originally the surgery exact triangle is only proved in characteristics two (c.f. [KM07,Section 42]). However, it is worth mentioning that recently the surgery exact triangle in characteristic zero was study by Freeman [Fre], so it might be possible to extend the discussion to F " Q or C for monopole theory. It is also worth mentioning that in [KM10b], when Kronheimer and Mrowka first introduced sutured monopole homology, they worked only with Z coefficients. The case of Z 2 coefficients was later verified by Sivek [Siv12].
The canonical Z 2 -grading for monopole theory was discussed by Kronheimer and Mrowka [KM07, Section 25.4]. When considering cobordisms of connected 3-manifolds, the degree formula (2.2) is the same as the formula in [KM07,Definition 25.4.1].
Heegaard Floer theory. The H-homology and the H-cobordism map are HF`pY q and HF`pW q (c.f. [OS04d]). Similar to monopole theory, we will use other versions of Heegaard Floer homology HF´, HF 8 , HF´, HF 8 for the proof of the Floer's excision theorem. See Section 3 for details.
The coefficient field is F " F 2 . This is because we have to use the naturality results in [JTZ18], which works only for F 2 . Originally, to obtain a p3`1q-TQFT, we should consider the graph cobordism category Cob 3`1 Γ (c.f. [Zem19]) rather than Cob 3`1 . However, after modifying the naturality results in Section 3, we can show the Floer homology and and the cobordism map are independent of the choice of basepoints and graphs.
For characteristic zero, the naturality results for closed 3-manifolds were obtained by Gartner in [Gar19]. However, the naturality results for cobordisms are still under working. Hence we choose to focus on characteristics two. Similar to monopole theory, the Z-grading in Axiom (A1) is induced by xc 1 psq, αy for s P Spin c pY q and α P H 2 pY q.
There are many ways to fix the Z 2 -grading for Heegaard Floer theory. See [OS04c, Section 10.4] and [FJR09, Section 2.4]. However, we can arrange the canonical Z 2 -grading to be the same as those for instanton theory and monopole theory. This is possible because the degree formula (2.2) only depends on algebraic-topological information of cobordisms and 3-manifolds.

Formal sutured homology of balanced sutured manifolds.
In [KM10b], Kronheimer and Mrowka constructed sutured monopole homology SHM and sutured instanton homology SHI by considering closures of balanced sutured manifolds. The discussion and construction in this subsection are based on [KM10b,BS15] except for the proof of Proposition 2.11.
Definition 2.7 ([Juh06, Definition 2.2]). A balanced sutured manifold pM, γq consists of a compact 3-manifold M with non-empty boundary together with a closed 1-submanifold γ on BM . Let Apγq " r´1, 1sˆγ be an annular neighborhood of γ Ă BM and let Rpγq " BM zintpApγqq, such that they satisfy the following properties.
(1) Neither M nor Rpγq has a closed component.
(2) If BApγq "´BRpγq is oriented in the same way as γ, then we require this orientation of BRpγq induces the orientation on Rpγq, which is called the canonical orientation.
(3) Let R`pγq be the part of Rpγq for which the canonical orientation coincides with the induced orientation on BM from M , and let R´pγq " RpγqzR`pγq. We require that χpR`pγqq " χpR´pγqq. If γ is clear in the contents, we simply write R˘" R˘pγq, respectively.
Definition 2.8 ( [KM10b]). Suppose pM, γq is a balanced sutured manifold. Let T be a connected compact oriented surface such that the numbers of components of BT and γ are the same. Let the preclosure Ă M of pM, γq be The boundary of Ă M consists of two components r R`" R`pγq Y t1uˆT and r R´" R´pγq Y t´1uˆT.
Let h : r R`-Ý Ñ r R´be a diffeomorphism which reverses the boundary orientations (i.e. preserves the canonical orientations). Let Y be the 3-manifold obtained from Ă M by gluing r R`to r R´by h and let R be the image of r R`and r R´in Y . The pair pY, Rq is called a closure of pM, γq. The genus of R is called the genus of the closure pY, Rq. For a closure pY, Rq with gpRq ě 2, define HpY |Rq :" HpY, rRs, 2gpRq´2q.
Remark 2.9. For instanton theory, we also choose a point p on T and choose a diffeomorphism h such that hpt1uˆpq " t´1uˆp. The image of r´1, 1sˆp in Y becomes a 1-cycle ω and we have ω¨R " 1. We use pY, R, ωq for the definition of a closure. We do not mention this subtlety later since everything works well under this modification.
Suppose pY 1 , R 1 q and pY 2 , R 2 q are two closures of pM, γq of the same genus. We now construct a canonical map Φ 12 : HpY 1 |R 1 q Ñ HpY 2 |R 2 q.
Note that Y 2 can be obtained from Y 1 as follows. There exists an orientation preserving diffeomorphism h 12 : R 1 Ñ R 1 so that if we cut Y 1 open along R 1 and reglue using h 12 , then we obtain a new 3-manifold Y 1 1 together with the surface R 1 It is straightforward to check HpX φ q : HpY 1 1 |R 1 1 q Ñ HpY 2 |R 2 q is an isomorphism. We can regard h 12 as a composition of Dehn twists along curves on R 1 : Here e i P t˘1u, where e 1 " 1 means a positive Dehn twist, and e 1 "´1 means a negative Dehn twist. Suppose N " ti | e i "´1u and P " ti | e i " 1u.
Note that the resulting 3-manifold of cutting Y 1 open along R 1 and regluing by D ei αi is the same as the resulting 3-manifold of performing a p´e i q-surgery along α i Ă R 1 Ă Y 1 . We take a neighborhood N pR 1 q of R 1 Ă Y 1 , and choose an identification N pR 1 q " r´1, 1sˆR 1 . Picḱ 1 ă t 1 ă¨¨¨ă t n ă 1 so that t i ‰ 0 for i " 1, . . . , n, and isotope α i to the level tt i uˆR 1 Ă N pR 1 q Ă Y 1 . Let Y P be the 3-manifold obtained from Y 1 by performing p´1q-surgeries along α i for all i P P . There is a natural cobordism X P from Y 1 to Y P by attaching framed 4-dimensional 2-handles to the product r0, 1sˆY 1 along α iˆt 1u. Furthermore, the manifold Y P can also be obtained from Y 1 1 by performing p´1q-surgeries along α i for all i P N . Hence there is a similar cobordism X N from Y 1 1 to Y P . Since t i ‰ 0, the surface R 1 " t0uˆR 1 survives in all surgeries. Let R P Ă Y P be the corresponding surface.
Remark 2.12. Proposition 2.11 restates [BS15, Lemma 4.9]. However, the proof in that paper involves a non-vanishing result for minimal Lefschetz fibrations. See [BS15,Proposition B.1]. Yet this non-vanishing result is not covered by Axioms (A1), (A2), and (A3), so we present an alternative proof of Proposition 2.11 based on surgery exact triangles from Axiom (A2). Also, it is worth mentioning that Baldwin and Sivek worked with Z coefficients for monopole theory in [BS15], while we work with Z 2 coefficients. The choice of coefficients matters since the existing proof of the surgery exact triangle in monopole theory is only carried out in characteristics two.
Proof of Proposition 2.11. The cobordisms X P and X N are constructed similarly, so we only prove X P is an isomorphism. Furthermore, we can assume that P has only one element α 1 . If it has more elements, then X P is simply the composition of cobordisms associated to single Dehn surgeries. With this assumption, the manifold Y P is obtained from Y 1 by performing a p´1q-surgery along α 1 . Let Y 0 be obtained from Y 1 by performing a 0-surgery along α 1 , and R 1 survives to become R 0 Ă Y 0 . Then we have an exact triangle by Axioms (A1-7) and (A2): To show that HpX P q is an isomorphism, it suffices to show that HpY 0 |R 0 q " 0. Indeed, since Y 0 is obtained from a 0-surgery along α 0 , and α 0 can be isotoped to be a simple closed curve on R 1 , the surface R 0 Ă Y 0 is compressible. Hence HpY 0 |R 0 q " 0 by the the adjunction inequality in Axiom (A1-4).
With Proposition 2.11 settled down, the rest of the argument in [BS15] can be applied to our setup verbatim, and we have the following theorem.
Theorem 2.13 ( [BS15]). Suppose pM, γq is a balanced sutured manifold and pY 1 , R 1 q and pY 2 , R 2 q are two closures of the same genus. Then the isomorphism Φ 12 : HpY 1 |R 1 q Ñ HpY 2 |R 2 q defined in Definition 2.10 satisfies the following properties.
(1) The map Φ 12 is well-defined up to multiplication by a unit in F. .
" means the equation holds up to multiplication by a unit in F.
(3) If there is a third closure pY 3 , R 3 q of the same genus, then we have Remark 2.14. In Baldwin and Sivek's original work, the requirement that the two closures have the same genus could be dropped, at the cost of involving local coefficient systems. However, up to the authors' knowledge, the discussion for the naturality of Heegaard Floer theory has not been carried out with local coefficients. Since it is enough to work with closures of a large and fixed closure in the current paper, we choose not to discuss on the local coefficients.
Definition 2.15 ([JTZ18, BS15]). A projectively transitive system of vector spaces over a field F consists of (1) a set A and collection of vector spaces tV α u αPA over F, (2) a collection of linear maps tg α β u α,βPA well-defined up to multiplication by a unit in F such that (a) g α β is an isomorphism from V α to V β for any α, β P A, called a canonical map, " g α γ for any α, β, γ P A. A morphism of projectively transitive systems of vector spaces over a field F from pA, tV α u, tg α β uq to pB, tU γ u, th γ δ uq is a collection of maps tf α γ u αPA,γPB such that (1) f α γ is a linear map from V α to U γ well-defined up to multiplication by a unit in F for any α P A and γ P B, " h γ δ˝f α γ for any α, β P A and γ, δ P B.
A transitive system of vector spaces over a field F if it is a projectively transitive system and all equations with . " are replaced by ones with ". A morphism of transitive systems of vector spaces over a field F is defined similarly.
We can replace vector spaces by groups or chain complexs of vector spaces and define the projectively transitive system and the transitive system similarly.
Remark 2.16. A transitive system of vector spaces pA, tV α u, tg α β uq over a field F canonically defines an actual vector space over F where v α " v β if and only if g α β pv α q " v β for any v α P V α and v β P V β . A morphism of transitive systems of vector spaces canonically defines an linear map between corresponding actual vector spaces.
Convention. If F " F 2 , a projectively transitive system over F is simply a transitive system since F 2 has only one unit. In this case, we do not distinguish the projectively transitive system, the transitive system and the corresponding actual vector space. For a general field F, the morphisms between projectively transitive systems are also called maps.
Definition 2.17. Suppose H is a p3`1q-TQFT satisfying Axioms (A1) and (A2), and pM, γq is a balanced sutured manifold, the formal sutured homology SH g pM, γq is the projectively transitive system consisting of (1) the H-homology HpY |Rq for closures pY, Rq of pM, γq with a fixed and large enough genus g.
(2) the canonical maps Φ between H-homologies as in Definition 2.10.
Convention. Through out the paper, when discussing formal sutured homology, we will pre-fix a large enough genus. So we omit it from the notation and write simply SHpM, γq instead of SH g pM, γq.
Remark 2.18. When H also satisfies Axiom (A3), since Φ is constructed by cobordism maps and their inverses, it is homogeneous with respect to the Z 2 -grading from Axiom (A3). Then there exists an induced relative Z 2 -grading on SHpM, γq.
In [BS16a,BS18], Baldwin and Sivek proved the bypass exact triangle for sutured monopole homology and sutured instanton homology. Their proof can be exported to our setup.
Theorem 2.19 ([BS16a, Theorem 5.2] and [BS18, Theorem 1.20]). Suppose pM, γ 1 q, pM, γ 2 q, pM, γ 3 q are three balanced sutured manifold such that the underlying 3-manifold is the same, and the sutures γ 1 , γ 2 , and γ 3 only differ in a disk as depicted in Figure 1. Then there exists an exact triangle Moreover, the maps ψ i are induced by cobordisms, hence are homogeneous with respect to the relative Z 2 -grading on SHpM, γ i q. We un-package the proof of Theorem 2.19 for later convenience.
(a) For i " 1, 2, 3, we have ζ i X intpM q " H and ζ i can be isotoped to be disjoint from R.
(b) If we perform a suitable Dehn surgery along ζ 1 , then we obtain a closure pY 2 , Rq of p´M,´γ 2 q. If we perform a suitable Dehn surgery in Y 2 along ζ 2 , then we obtain a closure pY 3 , Rq of p´M,´γ 3 q. If we perform a suitable Dehn surgery in Y 3 along ζ 3 , then we obtain the closure pY 1 , Rq of p´M,´γ 1 q again. (c) The maps ψ 1 , ψ 2 , and ψ 3 are induced the cobordism associated to Dehn surgeries along ζ 1 , ζ 2 , and ζ 3 , respectively. (3) There are two curves η 1 and η 2 on R, so that if we perform p´1q-surgeries on both of them, with respect to the surface framings from R, then the surgeries along ζ 1 , ζ 2 , and ζ 3 as stated in (b) lead to an exact triangle as in Axiom (A2).

Gradings on formal sutured homology.
Suppose pM, γq is a balanced sutured manifold and S Ă pM, γq is a properly embedded surface in M . If S satisfies some admissible conditions, the first author [Li19] constructed a Z-grading on SHM pM, γq and SHIpM, γq. In this subsection, we adapt the construction to formal sutured homology SHpM, γq. (1) Every boundary component of S intersects γ transversely and nontrivially.
(2) 1 2 |S X γ|´χpSq is an even integer. Recall the construction of a closure of pM, γq in Definition 2.8. Let T be a connected compact oriented surface of large enough genus and BT -´γ. Then we take Suppose n " 1 2 |BS X γ| and BS X γ " tp 1 , . . . , p 2n u. Definition 2.22 ([Li19]). A pairing P of size n is a collection of n couples P " tpi 1 , j 1 q, . . . , pi n , j n qu such that the followings hold.
(2) For any k P t1, . . . , nu, the points p i k and p j k are positive and negative, respectively, as intersection points of oriented curves BS and γ on BM .
Given a pairing P of size n, and assuming that gpT q is large enough, we can extend S to a properly embedded surface in Ă M as follows. Let α 1 ,. . . ,α n be pairwise disjoint properly embedded arcs on T such that the followings hold.
Given such α 1 , . . . , α n , take Then r S P is a properly embedded surface inside Ă M .
Definition 2.23. A pairing P is called balanced if r S P X r R`and r S P X r R´have the same number of components.
For any balanced pairing P, we can pick an orientation preserving diffeomorphism h : r R`-Ý Ñ r Rś o that hp r S P X r R`q " r S P X r R´. Thus, we obtain a closed oriented surfaceS P Ă Y in the closure pY, Rq induced by h. Define SHpM, γ, S, iq :" HpY, prRs, rS P sq, p2gpRq´2, 2iqq.
Theorem 2.24. Given an admissible surface S in a balanced sutured manifold pM, γq, the decomposition is independent of all the choices made in the construction and hence is well-defined.
Remark 2.25. As mentioned in the convention after Definition 2.17, when writing SH, we actually mean SH g for some large and fixed integer g. This means that all closures involved have the same genus g.
Proof of Theorem 2.24. The decomposition follows from Axioms (A1-1) and (A1-7). This gives a Z-grading on SHpM, γq. To show that this grading is well-defined, we need to show that it is independent of the following three types of choices: (1) the choice of the balanced pairing P, (2) the choice of arcs α 1 ,. . . ,α n with fixed endpoints, (3) the choice of the diffeomorphism h. In [Li19, Section 3.1], the grading has been shown to be independent of the choices of type (2) and (3). The proof involves only Axioms (A1) and (A2) and hence can be applied to our current setup. However, the original argument for choices of type (1) in [Li19, Section 3.3] involves closures of different genus, which is beyond the scope of our current paper as mentioned in Remark 2.14. Hence, we provide an alternative proof here. For the moment let us write the grading as SHpM, γ, S, P, iq to emphasize that the grading a priori depends on the choice of the balanced pairing. Theorem 2.24 then follows from the following proposition.
To relate two different pairings, in [Li19], the author introduced the following operation.
Definition 2.27. Suppose P is a pairing of size n and α 1 , . . . , α n are related arcs. Suppose k, l P t1, . . . , nu are two indices so that the followings hold.
(1) The arcs t1uˆα i k and t1uˆα i l belong to different components of r S P X R`.
(2) The arcs t´1uˆα i k and t´1uˆα i l belong to different components of r S P X R´. Then we can construct another pairing The operation of replacing P by P 1 is called a cut and glue operation. Lemma 2.29. Suppose P and P 1 are two balanced pairings that are related by a cut and glue operation, then for any i P Z, we have SHpM, γ, S, P, iq " SHpM, γ, S, P 1 , iq.
Proof. Suppose k and l are the indices involved in the operation. From the first part of the proof of Theorem 2.24, we can freely make choices of type (2) and (3). Hence we can assume that there is a disk D Ă intpT q so that α k and α l intersects D in two arcs as depicted in Figure 2. Suppose D`" t1uˆD Ă R`and D´" t´1uˆD Ă R´.
We can choose an orientation preserving diffeomorphism h : Let pY, Rq be the corresponding closure of pM, γq and s S P be the closed surface defining the grading SHpM, γ, S, Pq. Let β k " α k X D and β l " α l X D. It is straightforward to check that if we remove the two arcs β k and β l from D Ă T , and glue back two new arcs β 1 k and β 1 l as shown in the middle subfigure of Figure 2, then we obtain two new properly embedded arcs α 1 k and α 1 l on T so that then we obtain the surface s S P 1 Ă Y that gives rise to the grading SHpM, γ, S, P 1 q. The lemma then follows from the fact Figure 2. The disk D, the arcs β k , β l , β 1 k , β 1 l , and the surface U .
The equality (2.5) can be proved by constructing an explicit cobordism in Yˆr0, 1s from s S P Ă Yˆt0u to s S P 1 Ă Yˆt1u: in the product pYˆr0, 1s, s S Pˆr 0, 1sq, we can remove and glue back S 1ˆU Ă Dˆr0, 1s, where U Ă Dˆr0, 1s is the surface shown in the right subfigure of Figure 2.
Proof of Proposition 2.26. It follows immediately from Theorem 2.28 and Lemma 2.29.
Having constructed the grading, the rest of the arguments in [Li19, Section 3.3] can be applied to our current setup verbatim. Hence we have the following. Li18]). Suppose pM, γq is a balanced sutured manifold and S Ă pM, γq is an admissible surface. Then there is a Z-grading on SHpM, γq induced by S , which we write as This decomposition satisfies the following properties.
(5) For any i P Z, we have Proof. Term (1) comes from the adjunction inequality in (A1-4). Term (2)  Definition 2.33. Suppose pM, γq is a balanced sutured manifold. It is called a homology product if H 1 pM, R`pγqq " 0 and H 1 pM, R´pγqq " 0. It is called a product sutured manifold if where Σ is a compact surface with boundary.
If S Ă pM, γq is not admissible, then we can perform an isotopy on S to make it admissible.
Definition 2.35. Suppose pM, γq is a balanced sutured manifold, and S is a properly embedded surface. A stabilization of S is a surface S 1 obtained from S by isotopy in the following sense. This isotopy creates a new pair of intersection points: We require that there are arcs α Ă BS 1 and β Ă γ, oriented in the same way as BS 1 and γ, respectively, and the followings hold.
(2) α and β cobound a disk D with intpDq X pγ Y BS 1 q " H. The stabilization is called negative if BD is the union of α and β as an oriented curve. It is called positive if BD " p´αq Y β. See Figure 3. We denote by S˘k the surface obtained from S by performing k positive or negative stabilizations, repsectively. The following lemma is straightforward.
Lemma 2.36. Suppose pM, γq is a balanced sutured manifold, and S is a properly embedded surface. Suppose S`and S´are obtained from S by performing a positive and a negative stabilization, respectively. Then we have the following.
(2) If we decompose pM, γq along S´, then the resulting balanced sutured manifold pM 1 , γ 1 q is not taut, as R˘pγ 1 q both become compressible.
Remark 2.37. The definition of stabilizations of a surface depends on the orientations of the suture and the surface. If we reverse the orientation of the suture or the surface, then positive and negative stabilizations switch between each other.
One can also relate the gradings associated to different stabilizations of a fixed surface. The proof for SHM and SHI in [Li19,Wan20] can be adapted to our setup as well.
Theorem 2.38 ([Li19, Proposition 4.3] and [Wan20, Proposition 4.17]). Suppose pM, γq is a balanced sutured manifold and S is a properly embedded surface in M that intersects γ transversely. Suppose all the stabilizations mentioned below are performed on a distinguished boundary component of S. Then, for any p, k, l P Z such that the stabilized surfaces S p and S p`2k are both admissible, we have SHpM, γ, S p , lq " SHpM, γ, S p`2k , l`kq.
Note that S p is a stabilization of S as introduced in Definition 2.35, and, in particular, S 0 " S.
If we have multiple admissible surfaces, then they together induce a multi-grading. This is proved for SHM and SHI by Ghosh and the first author [GL19]. The proof can be adapted to our case without essential changes.
Theorem 2.40 ([GL19, Theorem 1.12]). Suppose pM, γq is a balanced sutured manifold and α P H 2 pM, BM q is a nontrivial homology class. Suppose S 1 and S 2 are two admissible surfaces in pM, γq such that Then, there exists a constant C so that Based on the relative Z 2 -grading from Remark 2.18 and the Z n -grading from Theorem 2.39, we can define graded Euler characteristic of formal sutured homology.
Remark 2.42. It can be shown by Theorem 2.40 that the definition of graded Euler characteristic is independent of the choice of S 1 , . . . , S n if we regard it as an element in ZrHs{˘H. If the admissible surfaces S 1 , . . . , S n and a particular closure of pM, γq is fixed, then the ambiguity of˘H can be removed.
Proposition 2.43. Suppose pM, γq is a balanced sutured manifold and S Ă pM, γq is an admissible surface. Suppose the disk as in Figure 1, where we perform the bypass change, is disjoint from BS. Let γ 2 and γ 3 be the resulting two sutures. Then all the maps in the bypass exact triangle (2.4) are grading preserving, i.e., for any i P Z, we have an exact triangle

Heegaard Floer homology and the graph TQFT
In this section, we discuss the modification of Heegaard Floer theory to make it suitable to formal sutured homology.
In this subsection and the next subsection, we provide an overview of the graph TQFT for Heegaard Floer theory, constructed by Zemke [Zem19] (see also [HMZ18,Zem18]), and list some properties which are relevant to proofs in the third subsection about the Floer's excision theorem.
Definition 3.1. A multi-pointed 3-manifold is a pair pY, wq consisting of a closed, oriented 3-manifold Y (not necessarily connected), together with a finite collection of basepoints w Ă Y , such that each component of Y contains at least one basepoint.
Given two multi-pointed 3-manifolds pY 0 , w 0 q and pY 1 , w 1 q, a ribbon graph cobordism from pY 0 , w 0 q to pY 1 , w 2 q is a pair pW, Γq satisfying the following conditions.
(1) W is a cobordism from Y 0 to Y 1 .
(2) Γ is an embedded graph in W such that Γ X Y i " w i for i " 0, 1. Furthermore, each point of w i has valence 1 in Γ.
(3) Γ has finitely many edges and vertices, and no vertices of valence 0.
(4) The embedding of Γ is smooth on each edge. (5) Γ is decorated with a formal ribbon structure, i.e., a formal choice of cyclic ordering of the edges adjacent to each vertex.
Definition 3.2. A restricted graph is a graph whose vertices have valence either 1 or 2. A ribbon graph cobordism pW, Γq from pY 1 , w 1 q to pY 2 , w 2 q is called a restricted graph cobordism if Γ is restricted (so the cyclic ordering is unique) and any component of Γ does not connect two basepoints of the same manifold Y i for i " 1, 2. A multi-pointed Heegaard diagram H " pΣ, α, β, wq for pY, wq is a tuple satisfying the following conditions.
The chain complex CF´pHq is a free F 2 rU w s-module generated by intersection points x P T α XT β . Define CF 8 pHq :" CF´pHq b F2rUws F 2 rU w , U´1 w s and CF`pHq :" CF 8 pHq{CF´pHq.
To construct a differential on CF´pHq, suppose H satisfies some extra admissibility conditions if b 1 pY q ą 0 (c.f. [Zem19, Section 4.7]). Let pJ s q sPr0,1s be an auxiliary path of almost complex structures on Sym n Σ and let π 2 px, yq be the set of homology classes of Whitney disks connecting intersection points x and y (c.f. [OS08, Section 3.4]). For φ P π 2 px, yq, let M Js pφq be the moduli space of J s -holomorphic maps u : r0, 1sˆR Ñ Sym n Σ which represent φ. The moduli space M Js pφq has a natural action of R, corresponding to reparametrization of the source. We write x M Js pφq :" M Js pψq{R.
For φ P π 2 px, yq, let µpφq be the expected dimension of M Js pφq for generic J s and let n wi pφq be the algebraic intersection number of tw i uˆSym n´1 Σ and any representative of φ. Define n w pφq :" pn w1 pφq, . . . , n wm pφqq.
For a generic path J s , define the differential on CF´pHq by extended linearly over F 2 rU w s. The differential B Js can be extended on CF 8 pHq and CF`pHq by tensoring with the identity map. For a disconnected multi-pointed 3-manifold pY, wq " pY 1 , w 1 q \ pY 2 , w 2 q, where Y i is connected for i " 1, 2, suppose H i is an admissible multi-pointed Heegaard diagram of Y i and suppose J si are corresponding generic paths of almost complex structures. For˝P t8,`,´u, let the chain complex associated to pY, wq be (3.1) pCF˝pH 1 \ H 2 q, B Js q :" pCF˝pH 1 q, B Js 1 q b F2 pCF˝pH 2 q, B Js 2 q.
Remark 3.5. In Zemke's original construction [Zem19], one should choose colors for basepoints and graphs to achieve the functoriality of TQFT. Here we implicitly choose different colors for all basepoints so that we have the following relation on the homology level (3.2) HpCF˝pH 1 \ H 2 q, B Js q " HpCF˝pH 1 q, B Js 1 q b F2 HpCF˝pH 1 q, B Js 2 q.
Since we will only consider restricted graph cobordisms, the choice of colors satisfies the functoriality. Note that in the construction of [HMZ18,Zem18], the colors of all basepoints are the same and all U wi are identified as U , so (3.1) and (3.2) are not true in those cases.
The chain homotopy type of pCF˝pHq, B Js q is independent of the choice of the admissible diagram H and the generic path J s . Indeed, we have the following theorem about naturality.
Moreover, similar results hold for CF 8 and CF`.
Since all chain complexes discussed above can be decomposed into spin c structures (c.f. [OS04d, Section 2.6]), we have the following definition.
Definition 3.7. Suppose pY, wq is a multi-pointed 3-manifold and s P Spin c pY q. For˝P t8,`,´u, define CF˝pY, w, sq to be the transitive system of chain complexes with canonical maps from Theorem 3.6, with respect to s, and define HF˝pY, w, sq to be the induced transitive system of homology groups.
For later use, we also define the completions of the chain complexes.
Definition 3.8. Let F 2 rrU w ss be the ring of formal power series of U w . For˝P t8,`,´u, define CF˝pY, w, sq :" CF˝pY, w, sq b F2rUws F 2 rrU w ss.
Let HF˝pY, w, sq be the induced homology groups.
Convention. When omitting the module structure, we have CF`pY, w, sq " CF`pY, w, sq. Hence we do not distinguish these groups.
The advantage of the completions is that we have the following proposition. Then the boundary map in the following long exact sequence induces a canonical isomorphism between HF´pY, w, sq and HF`pY, w, sq for any nontorsion spin c structure s. Proposition 3.10. From the short exact sequence 0 Ñ CF´pY, w, sq Ñ CF 8 pY, w, sq Ñ CF`pY, w, sq Ñ 0, we have a long exact sequencë¨¨Ñ HF´pY, w, sq Ñ HF 8 pY, w, sq Ñ HF`pY, w, sq Ñ¨¨D efinition 3.11. Suppose pY, wq is a multi-pointed 3-manifold and s P Spin c pY q is a nontorsion spin c structure. We write HF pY, w, sq " HF red pY, w, sq :" HF`pY, w, sq -HF´pY, w, sq.
Theorem 3.12 ([Zem19, Theorem A]). Suppose pW, Γq : pY 0 , w 0 q Ñ pY 1 , w 1 q is a ribbon graph cobordism and s P Spin c pW q. Then there are two chain maps F A W,Γ,s , F B W,Γ,s : CF´pY 0 , w 0 , s| Y0 q Ñ CF´pY 1 , w 1 , s| Y1 q, which is a diffeomorphism invariant of pW, Γq, up to F 2 rU w s-equivariant chain homotopy.
Proposition 3.13 ([Zem19, Theorem C]). Suppose that pW, Γq is a ribbon graph cobordism which decomposes as a composition pW, Γq " pW 2 , Γ 2 q Y pW 1 , Γ 1 q. If s 1 and s 2 are spin c structures on W 1 and W 2 , respectively, then A similar relation holds for F B W,Γ,s . Convention. Since we will only consider restricted graph cobordisms, the map F A W,Γ,s is chain homotopic to F B W,Γ,s . Hence we write CF´pW, Γ, sq for the chain map and HF´pW, Γ, sq for the induced map on the homology group. If Γ and s are specified, we write CF´pW q and HF´pW q for simplicity, respectively. The chain maps on CF 8 , CF`, CF´, CF 8 are obtained by tensoring with the identity maps, respectively. We use similar notations for these chain maps and the induced maps on homology groups. All maps are called cobordism maps.
For CF´, the cobordism map is defined by the composition of the following maps.
‚ For 4-dimensional 1-, 2-, and 3-handle attachments away from the basepoints, we use the maps defined by Ozsváth and Szabó [OS06]. ‚ For 4-dimensional 0-and 4-handle attachments, or equivalently adding and removing a copy of S 3 with a single basepoint, respectively, we use the maps defined by the canonical isomorphism from the tensor product with CF´pS 3 , w 0 q -F 2 rU 0 s. ‚ For a ribbon graph cobordism pYˆr0, 1s, Γq, we project the graph into Y and use the graph action map defined in [Zem19, Section 7].
Remark 3.14. For 4-dimensional 1-, 2-, and 3-handle attachments, Ozsváth and Szabó's original construction was for connected cobordisms between connected 3-manifolds. Zemke [Zem19, Section 8] extended the construction to cobordisms between possibly disconnected 3-manifolds. For 4dimensional 0-and 4-handle attachments, considering the coloring chain complex in [Zem19, Section 4.3], the isomorphism is indeed The graph action map is obtained by the composition of maps associated to elementary graphs. The construction involves free-stabilization maps Sw [Zem19, Section 6] and relative homology maps A λ [Zem19, Section 5], where Sw correspond to adding or removing a basepoint w and A λ correspond to a path λ between two basepoints. When considering restricted graph cobordisms, we only need maps associated to 1-, 2-, 3-handle attachments and free-stabilizations.
Let α 0 and β 0 be two simple closed curves on Σ bounding a disk containing w 0 and |α 0 X β 0 | " 2. Suppose θ`and θ´are the higher and the lower graded intersection points, respectively. Consider the Heegaard diagram H 1 " pΣ, α Y tα 0 u, β Y tβ 0 u, w Y tw 0 uq. See Figure 4. For appropriately chosen almost complex structures, define the free-stabilization maps Sw 0 by Sẁ 0 pxq " xˆθ`, Sẃ 0 pxˆθ´q " x and Sẃ 0 pxˆθ`q " 0. Remark 3.16. If we collapse BD to a point p 0 , we obtain a doubly-pointed diagram on S 2 with two curves. Hence H 1 can be considered as the connected sum of H and pS 2 , α 0 , β 0 , tw 0 , p 0 uq. (1) The maps Sw 0 commute with maps associated to 1-, 2-, and 3-handle attachments.
Remark 3.18. The free-stabilization maps can be regarded as restricted graph cobordisms with W " Yˆr0, 1s. The graphs are shown in Figure 5. Alternatively, we can regard them as compositions of maps associated to handle attachements. The map Sẁ 2 is obtained by first attaching a 0-handle with an arc whose one endpoint is on the boundary and the other is in the interior, and then attaching a product 1-handle away from basepoints; see Figure 5. The map Sẃ 2 is obtained by first attaching a 3-handle and then a 4-handle with an arc similarly.
Convention. All illustrations of cobordisms are from the top to the bottom. We can calculate the effect of free-stabilization maps on the homology explicitly.  CF´p´Y, w,sq -Hom F2 pCF`pY, w, sq, F 2 q.
The following proposition implies the choice of the basepoints is not important.
Proposition 3.21 ([Zem19, Corollary 14.19 and Corollary F]). Suppose pY, wq is a multi-pointed 3-manifold and w 1 P w. Then the π 1 pY, w 1 q action on HF´pY, wq is always the identity map.
Suppose pY 1 , w 1 q and pY 2 , w 2 q are two multi-pointed 3-manifolds with |w 1 | " |w 2 |. Suppose W is a cobordism from Y 1 to Y 2 and Γ Ă W is a collection of paths connecting w 1 and w 2 . Then the cobordism map HF´pW, Γq is independent of the choice of Γ. Moreover, if W " YˆI, then HF´pW, Γq is an isomorphism.
The similar results also hold for HF 8 , HF`, HF´, HF 8 .
From Corollary 3.20 and Proposition 3.21, we can define a transitive system of groups based on different choices of basepoints.
Definition 3.22. Suppose Y is a closed, oriented 3-manifold and w 1 , w 2 Ă Y are two collections of basepoints in Y . Let w 1 1 " w 1 zw 2 and w 1 2 " w 2 zw 1 . Define transition maps associated to pw 1 , w 2 q as Ψẃ 1Ñw2 :" where the products mean compositions. The order of maps is not important by the following lemma.
Lemma 3.23. Suppose Y is a closed, oriented 3-manifold and w 1 , w 2 , w 3 Ă Y are three collections of basepoints in Y . Suppose w is a basepoint in Y that is not in w i for i " 1, 2. Then the followings hold for transition maps.
(1) Ψw iÑwj is well-defined for i, j P t1, 2, 3u, i.e., the composition is independent of the order of maps.
Proof. This follows from term (1) of Proposition 3.17.
Theorem 3.25. Suppose Y is a closed, oriented 3-manifold. Then groups HF´pY, wq for all w Ă Y and transition maps Ψẃ 1Ñw2 for all w 1 , w 2 Ă Y form a transitive system, which is denoted by HF´pY q. Moreover, suppose pW, Γq is a restricted graph cobordism from pY 1 , w 1 q to pY 2 , w 2 q. Then HF´pW, Γq induces a well-defined map from HF´pY 1 q to HF´pY 2 q, which is independent of the choice of the restricted graph Γ and denoted by HF´pW q.
The similar arguments hold for HF´and HF`.
Proof. The well-definedness of HF´pY q and HF´pW, Γq follow from Lemma 3.23 and Lemma 3.24. The restricted graph cobordism is a composition of maps associated to 1-, 2-, 3-handle attachments and free-stabilizations. Note that free-stabilizations maps can be isomorphisms or zero maps on homology groups. Then the independence of Γ follows from Proposition 3.21 and above lemmas. The proofs for HF´and HF`are similar.
Remark 3.26. Groups and maps in Theorem 3.25 also split into spin c structures. Suppose s P Spin c pW q is a nontorsion spin c structure which restricts to nontorsion spin c structure s i on Y i for i " 1, 2. Then HF´pY i , s i q and HF`pY i , s i q are canonically identified by the boundary map in Proposition 3.10. Moreover, the maps HF´pW, sq and HF`pW, sq are the same under this identification. We write the map as HF pW, sq.

Floer's excision theorem.
Note that the proof of Theorem 2.13 in [BS15] and the proof of Theorem 2.24 in [Li19] both involves Floer's excision theorem in an essential way. In this subsection, we follow Kronheimer and Mrowka's idea in [KM10b, Section 3] to prove an excision theorem in Heegaard Floer theory. The proof depends essentially on the TQFT properties and Axioms (A1), so it works for a general TQFT satisfying the Axiom (A1). Though for Heegaard Floer theory, we need to modify the proof of the excision theorem to fit the settings of multi-basepoints and graph cobordisms.
Let Y be a closed, oriented 3-manifold, of either one or two components. In the latter case, let Y 1 and Y 2 be two components of Y . Let Σ 1 and Σ 2 be two closed, connected, oriented surfaces in Y with gpΣ 1 q " gpΣ 2 q. If Y has two components, suppose Σ i is a non-separating surface in Y i for i " 1, 2. If Y is connected, suppose Σ 1 and Σ 2 represent independent homology classes. In either case, let F " Σ 1 Y Σ 2 . Let h be an orientation-preserving diffeomorphism from Σ 1 to Σ 2 .
We construct a new manifold r Y as follows. Let Y 1 be obtained from Y by cutting along Σ. Then If Y has two components, then we have Y be obtained from Y 1 by gluing the boundary component Σ 1 to the boundary component´Σ 2 and gluing Σ 2 to´Σ 1 , using the diffeomorphism of h in both cases; see Figure 6 for the case that Y has two components. In either case, r Y is connected. Let r Σ i be the image of Σ i in r Y for i " 1, 2 and let r Definition 3.27. Suppose Y is a closed, oriented 3-manifold and F Ă Y is a closed, oriented surface. Let Let HF`pW, Γ|Gq, HF´pW, Γ|Gq and HF pW, Γ|Gq be defined similarly.
Remark 3.28. All spin c structures in Spin c pY |F q are nontorsion, so HF pY, sq is well-defined.
The following is the main theorem of this subsection.
Theorem 3.29 (Floer's excision theorem). Consider Y and r Y constructed as above. If gpΣ 1 q " gpΣ 2 q ě 2, then there is an isomorphism Moreover, this isomorphism and its inverse are induced by restricted graph cobordisms.
Before proving the main theorem, we introduce some lemmas parallel to results in monopole theory (c.f. [KM10b, Lemma 2.2, Proposition 2.5 and Lemma 4.7]) Lemma 3.30 ([Lek13, Corollary 17], see also [OS04a, Theorem 5.2]). Let Y Ñ S 1 be a fibred 3-manifold whose fibre F is a closed, connected, oriented surface with g " gpF q ě 2. Then there is a unique s 0 P Spin c pY |Rq so that HF pY, s 0 q ‰ 0. Moreover, we have HF pY |F q " HF pY, s 0 q -F 2 .
Lemma 3.31. Suppose Y " ΣˆS 1 such that Σ " Σˆt1u Ă Y is a closed, connected, oriented surface with gpΣq ě 2. Suppose w 0 P S 3 and w P Y are basepoints. Let W be obtained from ΣˆD 2 by removing a 4-ball, considered as a cobordism from S 3 to Y . Let Γ Ă W be any path connecting w 0 and w 1 . Then the map HF´pW, Γ|Σq : F 2 rrU 0 ss -HF´pS 3 , w 0 q Ñ HF pY |Σq -F 2 is nontrivial.
Proof. Suppose P is 2-dimensional pair of pants as shown in the left subfigure of Figure 7. Consider W 1 " ΣˆP as a cobordism from Y 1 \ Y 2 to Y 3 , where Y i -Y for i " 1, 2, 3. Suppose w 1 is another basepoint in Y . Let w i and w 1 i be images of w and w 1 in Y i for i " 1, 2, 3. Let Γ 1 Ă W 1 be a collection of two paths γ 1 and γ 2 , where γ 1 connects w 1 1 to w 1 3 and γ 2 connects w 2 to w 3 . Let pW 1 , Γ 1 q " pY 1ˆI , w 1ˆI q be the product cobordism. Suppose Σ i Ă Y i is the image of Σ Ă Y for i " 1, 2, 3. Consider the composition of the cobordism maps After filling the S 3 component by a 4-ball, or equivalently composing it with the map associated to a 0-handle attachment, we obtain the free-stabilization map Sẁ (c.f. Remark 3.18). By Corollary 3.20, the resulting map is an isomorphism HF pY 1 |Σ 1 q -HF pY 3 |Σ 3 q.
The proof about HF`p´W, Γ|Σq is similar. We replace W 1 by´W 1 and Sẁ by Sẃ in the above proof. See the right subfigure of Figure 7 for the illustration of the composition of the cobordisms.
Proof. By Lemma 3.30, we have HF pY |Σq -F 2 , so any nontrivial map induces the identity map. By Lemma 3.31, the composition is a nontrivial map. Remark 3.33. The proof of [KM10b, Proposition 2.5] involves some arguments about the trace map, while the setting about trace cobordism in [Zem18] is different from what we use here (c.f. Remark 3.5). Corollary 3.32 implies that we can replace the cobordism in the left subfigure of Figure 8 by the right subfigure. This is not true for a general coloring in [Zem19]. To apply this replacement, we should decompose cobordisms in [KM10b, Section 3.2] into compositions of cobordisms.
Now we start to prove the main theorem of this subsection. The basic idea is from [KM10b, Section 3.2].
Proof of Theorem 3.29.
Step 1. We construct a cobordism W from r Y to Y and a cobordismW from Y to r Y . Recall that Y 1 is obtained from Y by cutting along Σ 1 and Σ 2 and we have Suppose P 1 is a saddle surface, which can be regarded as a submanifold of a pair of pants with one boundary component on the top and two boundary components at the bottom. See Figure 9.
where λ 1 and λ 2 are two arcs in the top boundary component of the pair of pants, µ 1 and µ 2 are two arcs in the bottom boundary components of the pair of pants, and η i,j is the arc connecting λ i and µ j for i, j P t1, 2u.
Suppose Σ -Σ 1 -Σ 2 . Note that we have fixed a diffeomorphism h from Σ 1 to Σ 2 . Suppose h 1 is an orientation-preserving diffeomorphism from Σ to Σ 1 . Let W be the union where η 1,1ˆΣ is glued to Σ 1ˆI , η 2,1ˆΣ is glued to´Σ 1ˆI , η 2,2ˆΣ is glued to Σ 2ˆI , and η 1,2ˆΣ is glued to´Σ 2ˆI , using h 1 and h˝h 1 , respectively. Figure 9 illustrates the case that Y 1 has two components Y 1 1 and Y 1 2 . By the construction of r Y , the resulting manifold W is a cobordism from r Y to Y . The cobordismW is constructed similarly. Let P 2 be another saddle surface and letW be obtained by gluing P 2ˆΣ and Y 1ˆI as shown in Figure 9.
Step 2. For some restricted graph Γ and some surface G in W A "W Y r Y W , we show the cobordism map HF pW A , Γ|Gq :" HF`pW A , Γ|Gq " HF´pW A , Γ|Gq induce the identity map on HF pY |F q :" HF`pY |F q -HF´pY |F q.
We prove for the case that Y has two components Y 1 and Y 2 . The proof for the case that Y is connected is similar. The cobordism W A is shown in the right subfigure of Figure 9. Let w 1 and w 1 1 be two basepoints on Y 1 and let w 2 and w 1 2 be two basepoints on Y 2 . Moreover, we can suppose w i P Y 1 i for i " 1, 2. For i " 1, 2, let γ i be the union of images of w iˆI Ă Y 1 iˆI inW and W . Let γ 3 Ă W A be a path in the union of images of P 1ˆΣ and P 2ˆΣ connecting w 1 1 to w 1 2 , as shown in Figure 10. Let It is straighforward to check by definition that Γ A is restricted. Let w be a basepoint on ΣˆS 3 . We decompose the restricted graph cobordism pW A , Γq into four pieces W i as shown in Figure 10, where W 2 and W 3 are product cobordisms for pY 1 \ ΣˆS 1 \ Y 2 , tw 1 , w, w 2 uqq. Let By Remark 3.26, we know HF´pW A , Γ A |Gq and HF`pW A , Γ A |Gq induce the same map, so HF´pW i , Γ A X W i |Gq and HF`pW A , Γ A X W i |Gq also induce the same map. We assign HFt o W 1 and W 2 and HF`for W 3 and W 4 .  Note that we have By Corollary 3.32, we can replace the components of W 2 and W 3 corresponding to ΣˆS 1 by the cobordism in the left subfigure of Figure 8. After filling two S 3 boundaries by 4-balls, the resulting graph cobordism pW 1 A , Γ 1 A q is the composition of the graph cobordisms associated to Sẁ1 2 and Sẃ1 1 . By Corollary 3.20 and our assignments of the cobordism maps, we know that HF pW A , Γ A |Gq " HF pW 1 A , Γ 1 A |Gq is an isomorphism. By construction in Definition 3.22, we know that HF pW A , Γ A |Gq induces the identity map on HF pY |F q.
Step 3. For some restricted graph Γ 1 and some surface G 1 in W B " W Y YW , we show the cobordism map HF pW B , Γ 1 |G 1 q :" HF`pW B , Γ 1 |G 1 q " HF´pW B , Γ 1 |G 1 q induce the identity map on We prove for the case that Y has two components Y 1 and Y 2 . The proof for the case that Y is connected is similar. The cobordism W B is shown in the left subfigure of Figure 9. The proof is essentially the same as that in Step 2. However, the decomposition of W B is not obvious as before.
Let w 1 , w 2 , w 2 1 and w 2 2 be basepoints on r Y . Let γ 1 1 and γ 1 2 be defined similarly to γ 1 and γ 2 , respectively. Let γ 1 3 Ă W B be a path in the union of images of P 1ˆΣ and P 2ˆΣ connecting w 2 1 to w 2 2 , as shown in Figure 11. Let

It is straighforward to check by definition that Γ B is restricted.
Similar to the proof for W A , we decompose W B into four pieces W 1 i , where W 1 2 and W 1 3 are product cobordisms for p r Y \ ΣˆS 1 , tw 1 , w 2 , wuqq. In Figure 11 we only draw W 1 1 and W 1 4 . Let Similarly, We assign HF´to W 1 1 and W 1 2 and HF`for W 1 3 and W 1 4 . By similar argument to that in the proof for W A , we know that HF pW B , Γ B |G 1 q induces the identity map on HF p r Y | r F q.
Step 4. The results in Step 2 and Step 3 imply Then we know restricted graph cobordisms pW, Γ A X W q, pW , Γ A XW q, pW, Γ B X W q and pW , Γ B XW q with respect to G and G 1 induce isomorphisms HF pW q and HF pW q between HF pY |F q and HF p r Y | r F q, respectively.

Sutured Heegaard Floer homology.
In this subsection, we introduce two equvalent definitions of sutured Heegaard Floer homology. The first one, is due to Juhász [Juh06], based on balanced diagrams of balanced sutured manifolds. The other follows from the construction in Section 2.2, which is essentially due to Kronheimer and Mrowka [KM10b]. These definitions are denoted by SF H and SHF, respectively. The equivalence of these definitions was shown by Lekili [Lek13], Baldwin and Sivek [BS20b]. We will focus on the equality for graded Euler characteristics of two homologies. (1) Σ is a compact, oriented surface with boundary.
Suppose H " pΣ, α, βq is a balanced diagram with g " gpΣq and n " |α| " |β|. The chain complex SF CpHq is a free F 2 -module generated by intersection points x P T α X T β . Similar to the construction of CF´, for a generic path of almost complex structures J s on Sym n Σ, define the differential on SF CpHq by   Remark 3.40. By work of Kutluhan, Lee, and Taubes [KLT10], for any s P Spin c pY q, there is an isomorphism HF`pY, sq -HM˚pY, sq "HM ‚ pY, sq.
The last group is used to define SHM in [KM10b]. Following the discussion in Section 2.2, we can prove the naturality of SHF pM, γq based on the Floer's excision theorem. Let SHFpM, γq be the transitive system corresponding to SHF pM, γq.
(3) Suppose α 1 " α Y α 2 and β 1 " β Y β 2 . There exists an intersection point x 1 P T α 2 X T β 2 so that the map f : SF CpHq Ñ HF`pH 1 |Rq c Þ Ñ rcˆx 1 , 0s is a quasi-isomorphism. with respect to the grading associated to H and the Z 2 grading, up to a global grading shift.
Proof. It suffices to show the quasi-isomorphism in Theorem 3.41 respects spin c structures and Z 2 -gradings. Consider the Z 2 -gradings at first. Suppose c 1 and c 2 are two generators of SF CpHq. Note that the Z 2 -grading of c i is defined by the sign of the corresponding intersection point in T α X T β for i " 1, 2. For c iˆx1 , the Z 2 -grading is defined by mod 2 Maslov grading, which coincides with the sign of the corresponding intersection point in T α 1 X T β 1 . Thus, we have gr 2 pc 1 q´gr 2 pc 2 q " gr 2 pc 1ˆx1 q´gr 2 pc 2ˆx1 q, where gr 2 is the Z 2 -grading.
Then we consider spin c structures. Consider c i for i " 1, 2 as above. From [Juh06, Lemma 4.7], there is a one chain γ c1´γc2 such that spc 1 q´spc 2 q " PDprγ c1´γc2 sq, where sp¨q : T α X T β Ñ Spin c pM, BM q is defined in [Juh06, Definition 4.5], and PD : H 1 pM q Ñ H 2 pM, BM q is the Poincaré duality map.
From [OS04d, Lemma 2.19], we have s z pc 1ˆx1 q´s z pc 2ˆx1 q " PD 1 pi˚prγ c1´γc2 sqq, where s z p¨q : T α 1 X T β 1 Ñ Spin c pY q is defined in [OS04d, Section 2.6] and PD 1 : H 1 pY q Ñ H 2 pY q is the Poincaré duality map, and i˚: H 1 pM q Ñ H 1 pY q is the map induced by inclusion i : M Ñ Y . Hence we have c 1 ps z pc 1ˆx1 qq´c 1 ps z pc 2ˆx1 qq " 2PD 1 pi˚prγ c1´γc2 sqq.
Finally, the argument about graded Euler characteristics follows from definitions.

The graded Euler characteristic of formal sutured homology
In this section, we prove the graded Euler characteristic of formal sutured homology is independent of the choice of the Floer-type theory. Throughout this section, we assume that H is a Floer-type theory, i.e., it satisfies all three axioms (A1), (A2), and (A3). For simplicity, we say 'a property is independent of H' if a property is independent of the choice of the Floer-type theory. Suppose pM, γq is a balanced sutured manifold. If the admissible surfaces and the closure of pM, γq are fixed, then the graded Euler characteristic χ gr pSHpM, γqq in Definition 2.41 is considered as a well-defined element in ZrH 1 pM q{Torss, rather than ZrH 1 pM q{Torss{˘pH 1 pM q{Torsq; see Remark 2.42. Note that in this subsection, we avoid using H to denote H 1 pM q{Tors and the symbol H usually a handlebody.

Balanced sutured handlebodies.
In this subsection, we deal with Z n -gradings for a balanced sutured handlebody. We start with the following lemma about the sign ambiguity.
Lemma 4.1. Suppose pM, γq is a balanced sutured manifold, S Ă pM, γq is an admissible surface. Suppose pY 1 , R 1 q and pY 2 , R 2 q are two closures of pM, γq of the same genus so that S extends to closed surfacesS 1 andS 2 as in Subsection 2.3. If χ gr pHpY 1 |R 1 qq is already determined without the sign ambiguity, then χ gr pHpY 2 |R 2 qq is determined without the sign ambiguity from χ gr pHpY 1 |R 1 qq and the topological data of pY 1 , R 1 q and pY 2 , R 2 q.
From the proof of Theorem 2.24, the canonical map Φ 12 necessarily preserves the grading induced by S. From the construction of Φ 12 in Subsection 2.2, the canonical map is a composition of a few cobordism maps (or the inverse). Then the Z 2 -grading shift follows from Axiom (A3-3) .
Next, we consider gradings associated to admissible surfaces. To fix the ambiguity of H 1 pM q{Tors, we will fix the choices of admissible surfaces. For sutured handlebodies, we start with embedded disks.
Proposition 4.2. Suppose H is a genus g ą 0 handlebody and γ Ă BH is a closed oriented 1submanifold so that pH, γq is a balanced sutured manifold. Pick D 1 , . . . , D g a set of pairwise disjoint meridian disks in H so that rD 1 s, . . . , rD g s generate H 2 pH, BHq. Then for any fixed multi-grading i " pi 1 , . . . , i g q P Z g associated to D 1 , . . . , D g , the Euler characteristic χpSHp´H,´γ, iqq P Z{t˘1u depends only on pH, γq, D 1 , . . . , D g and i P Z g , and is independent of H. Furthermore, if a particular closure of p´H,´γq is fixed, then the sign ambiguity can be removed.
Proof. We fix the handlebody H and the set of disks D 1 , . . . , D g Ă H. For any suture γ on BH, define where |¨| denotes the number of points. We prove the proposition by induction on Ipγq. Since rγs " 0 P H 1 pBHq, we know |D j X γ| is always even for j " 1, . . . , g.
Note that an isotopy of γ can be understood as combinations of positive and negative stabilizations in the sense of Definition 2.35, and the grading shifting behavior under such isotopies (stabilizations) is described by Proposition 2.38, which is determined purely by topological data and is independent of H. Hence we can assume that the suture γ has already realized Ipγq.
First, if Ipγq ă 2g, then there exists a meridian disk D j with D j X γ " H. Then it follows from Theorem 2.32 that SHp´H,´γq " 0 since´H is irreducible while p´H,´γq is not taut. Hence for any multi-grading i P Z g , we have χpSHp´H,´γ, iqq " 0.
If Ipγq " 2g, then either there exists some integer j so that D j Xγ " H or for j " 1, . . . , g, we have |D j X γ| " 2. In the former case, we know that SHp´H,´γq " 0 and hence χpSHp´H,´γ, iqq " 0 for any multi-grading i P Z g . In the later case, we know that p´H,´γq is a product sutured manifold. It follows from Proposition 2.34 and Proposition 2.30 that SHp´H,´γq " SHp´H,´γ, 0q -F.

#˘1
i " 0 " p0, . . . , 0q 0 i P Z g zt0u Note that the ambiguity˘1 comes from the choice of the closure. If we choose a particular closure Y of p´H,´γq, then the Euler characteristic has no sign ambiguity. Since pH, γq is a product sutured manifold, there is a 'standard' closure pS 1ˆΣ , t1uˆΣq as in [KM10b]. By Axiom (A3-2), we have Then for any other closure pY, Rq, by Lemma 4.1 χ gr pSHpY |Rqq has no sign ambiguity. Now suppose we have proved that, for all γ so that Ipγq ă 2n, the Euler characteristic of SHp´H,´γ, iq, viewed as an element in Z{t˘1u, is independent of H, and that when we choose any fixed closure of p´H,´γq, the sign ambiguity can be removed. Next we deal with the case when Ipγq " 2n.
Note that we have dealt with the base case Ipγq ď 2g, so we can assume that n ě g`1. Hence, without loss of generality, we can assume that |D 1 X γ| ě 4. Within a neighborhood of BD 1 , the suture γ can be depicted as in Figure 12. We can pick the bypass arc α as shown in the same figure. From Proposition 2.43, for any multi-grading i P Z g , we have an exact triangle (4.1) SHp´H,´γ, iq Figure 12. The bypass arc α that reduces the intersection function I.
Note that the suture γ 1 and γ 2 are determined by the original suture γ and the bypass arc α, which are all topological data. From Figure 12, it is clear that Ipγ 1 q ď Ipγq´2 and Ipγ 2 q ď Ipγq´2.
Hence the inductive hypothesis applies, and we know that the Euler characteristics of SHp´H,´γ 2 , iq and SHp´H,´γ 1 , iq can be fixed independently of H. Note that the maps in the bypass exact triangle (4.1) are described by Proposition 2.20. Hence we conclude that the Euler characteristic of SHp´H,´γ, iq is also independent of H. Thus, we finish the proof by induction.
Next, we deal with gradings associated to general admissible surfaces. Proposition 4.3. Suppose H is a genus g handlebody, and S is a properly embedded surface in H. Suppose γ Ă BH is a suture so that pH, γq is a balanced sutured manifold and S is an admissible surface. Then the Euler characteristic χpSHp´H,´γ, S, jqq P Z{t˘1u depends only on pH, γq, S, and j P Z and is independent of H. Furthermore, if we fix a particular closure of p´H,´γq, then the sign ambiguity can also be removed.
Before proving the theorem, we need the following lemma.
Lemma 4.4. Suppose pM, γq is a balanced sutured manifold and S Ă pM, γq is a properly embedded admissible surface. Suppose α is a boundary component of S so that α bounds a disk D Ă BM and |α X γ| " 2. Let S 1 be the surface obtained by taking the union S Y D and then push D into the interior of M . Then for any i P Z, we have SHpM, γ, S, iq " SHpM, γ, S 1 , iq.
Proof. Push the interior of D into the interior of M and make D X S 1 " H. It is clear that rSs " rS 1 Y Ds P H 2 pM, BM q and BS " BpS 1 Y Dq.
In Subsection 2.3, when constructing the grading associated to S 1 Y D, we can pick a closure pY, Rq of pM, γq, so that S 1 and D extend to closed surfacesS 1 andD in Y , respectively. Since |BD X γ| " 2, we know thatD is a torus. Since BS " BpS 1 Y Dq, we know that S also extends to a closed surfacē S and from the fact that rSs " rS 1 Y Ds we know that rSs " rS 1 YDs " rS 1 s`rDs.
Proof of Proposition 4.3. It is a basic fact that the map B˚: H 2 pH, BHq Ñ H 1 pBHq is injective, and H 2 pH, BHq is generated by g meridian disks, which we fix as D 1 , . . . , D g . Hence we assume that rSs " a 1 rD 1 s`¨¨¨`a g rD g s P H 2 pH, BHq. Case 1. BS consists of only BD i , i.e.,

BS "
where Y ai BD i means the union of a i parallel copies of BD i .
Hence this case follows from Proposition 4.2. Case 2. BS contains some component that is not parallel to BD i for j " 1, . . . , g.
Step 1. We modify S and show that it suffices to deal with the case when S X D j " H for j " 1, . . . , g.
Note that impB˚q Ă H 1 pBHq is generated by rBD 1 s, . . . , rBD g s, so we have BS¨BD i " 0 for j " 1, . . . , g. Here¨denotes the algebraic intersection number of two oriented curves on BH. This means that for j " 1, . . . , g, the intersection points of BD i with BS can be divided into pairs. Suppose two intersection points of BD 1 with BS of opposite signs are adjacent to each other on BD 1 , as depicted in Figure 13. We can perform a cut and paste surgery along D 1 and S to obtain a new surface S 1 . From the same figure, it is clear that after isotopy, we can make Note that if we perform a cut and paste surgery along S 1 and´D 1 , we obtain another surface S 2 . From Figure 14 it is clear that BS 2 " BS Y θ, where θ is the union of some null-homotopic closed curves on BH. We can isotope S 2 to make each component of θ intersects the suture twice. Let Hence from Lemma 4.4 we know that SHp´H,´γ, S, jq " SHp´H, γ, S 4 , jq " SHp´H, γ, S 3 , jq " SHp´H,´γ, S 2 , j`jpS 2 , S 3 qq " à j1`j2"j`jpS2,S3q SHp´H,´γ, pD 1 , S 1 q, pj 1 , j 2 qq By Proposition 2.38, the shift jpS 2 , S 3 q depends on the isotopy from S 2 to S 3 , which is determined by the topological data and is independent of H. Hence we reduce the problem to understanding the Euler characteristic of SHp´H,´γq with multi-grading associated to pD 1 , S 1 q, with Repeating this argument, we finally reduce to the problem of understanding the Euler characteristic of SHp´H,´γq with multi-grading associated to pD 1 , . . . , D g , S g q, with BD i X BS g " H for j " 1, . . . , g.
Step 2. We modify S further to reduce to Case 1.
If every component of BS g is homotopically trivial, then we know that rS g s " 0 P H 2 pH, BHq, since the map H 2 pH, BHq Ñ H 1 pBHq is injective. We isotope each component of BS g by stabilization to make it intersect the suture γ twice and then cap it off by a disk. The resulting surface S g`1 is a homologically trivial closed surface in H, so SHp´H,´γq is totally supported at grading 0 with respect to S g`1 . The grading shift between S g and S g`1 can then be understood by Proposition 2.38, and is independent of H. Note that BHzpBD 1 Y¨¨¨Y BD g q is a 2g-punctured sphere, so BS is homotopically trivial when removing punctures on the sphere. If some component C of BS g is not null-homotopic, then C is obtained from some BD j by performing handle slides (or equivalently band sums) over BD 1 , . . . , BD g for some times.
If we isotope C to make it intersect some BD i twice and then apply the cut and paste surgery, the resulting curve is isotopic to the one obtained by performing a handle slide over BD i . Explicitly, in Figure 13, suppose two right endpoints of arcs in BS (the green arcs) are connected, then the right part of BS 1 is a trivial circle and the left part of BS 1 is obtained from BS by performing a handle slide over BD 1 . Thus, we can apply the cut and paste surgery for many times, which is equivalent to performing handle slides over BD 1 , . . . , BD g for some times. Finally, we reduce C to the curve isotopic to BD j . Then we reduce the problem to understanding the Euler characteristic of SHp´H,´γq with multi-grading associated to pD 1 , . . . , D g , S g`2 q, where S g`2 is a surface so that each component of BS g`2 is parallel to˘BD i for some i. Case 1 applies to S g`2 and we finish the proof.
Corollary 4.5. Suppose H is a handlebody and γ is a suture on BH so that pH, γq is a balanced sutured manifold. Suppose S 1 , . . . , S n are properly embedded admissible surfaces in pH, γq. Then the Euler characteristic χpSHp´H,´γ, pS 1 , . . . , S n q, pi 1 , . . . , i n qqq P Z{t˘1u depends only on pH, γq, S 1 , . . . , S n , and pi 1 , . . . , i n q P Z n , and is independent of H. Furthermore, if we fix a particular closure of p´H,´γq, then the sign ambiguity can also be removed.
Proof. The proof is similar to that for Proposition 4.3.

Gradings about contact 2-handle attachments.
In this subsection, we prove a technical proposition about the grading behavior for the map associated to contact 2-handle attachments.
Suppose M is a compact oriented 3-manifold with boundary, and S Ă M is a properly embedded surface. Suppose α Ă M is a properly embedded arc that intersects S transversely and Bα X BS " H. Let N " M zintpN pαqq, S N " S X N , and µ Ă BN be a meridian of α that is disjoint from S N . Let γ N be a suture on BN satisfies the following properties.
(2) If we attach a contact 2-handle along µ in the sense of [BS16a, Section 4.2], then we obtain a balanced sutured manifold pM, γ M q.
From [BS16a, Section 4.2], there is a map C µ : SHp´N,´γ N q Ñ SHp´M,´γ M q constructed as follows.
Push µ into the interior of N to become µ 1 . Suppose pN 0 , γ N,0 q is the manifold obtained from pN, γ N q by a 0-surgery along µ 1 with respect to the framing from BN . Equivalently, pN 0 , γ N,0 q can be obtained from pM, γ M q by attaching a 1-handle. Since µ 1 Ă intpN q, the construction of the closure of pN, γ N q does not affect µ 1 . Thus, we can construct a cobordism between closures of pN, γ N q and pN 0 , γ N,0 q by attaching a 4-dimensional 2-handle associated to the surgery on µ 1 . This cobordism induces a cobordism map It is shown in [BS16a, Section 4.2] (or also [KM10b, Section 6]) that attaching a product 1-handle does not change the closure, so there is an identification The main result of this subsection is the following proposition.
Proposition 4.6. Consider the setting as above. For any i P Z, we have Proof.
Step 1. We consider the grading behavior of the map C µ 1 for gradings associated to S N and S.
Since µ is disjoint from S, so we can also make µ 1 disjoint from S N " S X N . As a result, the surface S N survives in pN 0 , γ N,0 q. From Axiom (A1-7), the cobordism map associated to the 0-surgery along µ 1 preserves the grading associated to S N C µ 1 pSHp´N,´γ N , S N , iqq Ă SHp´N 0 ,´γ N,0 , S N , iq.
Step 2. We show ι : SHp´M,´γ M , S, iq " Ý Ñ SHp´N 0 ,´γ N,0 , S, iq. As discussed above, pN 0 , γ N,0 q is obtained from pM, γ M q by a product 1-handle attachment. This product 1-handle can be described explicitly as follows. In pN 0 , γ N,0 q, there is an annulus A bounded by µ and its push-off µ 1 . We can cap off µ 1 by the disk coming from the 0-surgery, and hence obtain a disk D with BD " µ. By assumption, we know that |BD X γ N,0 | " |µ X γ N | " 2. Hence D is a compressing disk that intersects the suture twice. If we perform a sutured manifold decomposition on pN 0 , γ N,0 q along D, it is straightforward to check the resulting balanced sutured manifold is pM, γ M q. However, in [Juh16], it is shown that decomposing along such a disk is the inverse operation of attaching a product 1-handle, and the disk is precisely the co-core of the product 1-handle. From this description, we can consider the product 1-handle attached to pM, γ M q as along two endpoints of α. Since Bα X BS " H, the surface S naturally becomes a properly embedded surface in pN 0 , γ N,0 q. From Axiom (A1-7), we know that the map ι preserves the gradings as claimed.
If S X α " H, then S " S N " S X N and the above argument is trivial. If S X α ‰ H, then S N is obtained from S by removing disks containing intersection points in α X S. Then BS N zBS consists of a few copies of meridians of α. For simplicity, we assume that there is only one copy of the meridian of α, i.e., BS N zBS " µ. The general case is similar to prove.
After performing the 0-surgery along µ 1 , we know that the surface S N Ă N 0 is compressible. Indeed, we can pick µ 2 Ă intpS N q parallel to µ Ă BS N . Then there is an annulus A 1 bounded by µ 2 and µ 1 , and we obtain a disk D 1 by capping µ 1 off by the disk coming from the 0-surgery. Performing a compression along the disk D 1 , we know that S N becomes the disjoint union of a disk D 2 and the surface S Ă N 0 . Note BD 2 is parallel to the disk D discussed above. Since BpD 2 Y Sq " BS N and rD 2 Y Ss " rS N s P H 2 pN 0 , BN 0 q, From (A1-6), we know that Since the disk D 2 intersects γ 1 N twice, from term (2) of Proposition 2.30, we know that SHp´N 0 ,´γ N,0 q " SHp´N 0 ,´γ N,0 , D 2 , 0q.
Hence we conclude that Remark 4.7. Theorem 4.6 is a generalization of [BLY20, Lemma 2.2], where α is a tangle and S N is an annulus.

General balanced sutured manifolds.
In this subsection, we prove the main theorem of this section, which is a restatement of the second part of Theorem 1.12.
Theorem 4.8. Suppose pM, γq is a balanced sutured manifold and tS 1 , . . . , S n u is a collection of properly embedded admissible surfaces. Then the Euler characteristic χpSHp´M,´γ, pS 1 , . . . , S g q, pi 1 , . . . , i n qqq depends only on pM, γq, S 1 , . . . , S n , and pi 1 , . . . , i n q P Z n , and is independent of H. Corollary 4.9. Suppose pM, γq is a balanced sutured manifold and suppose H " H 1 pM q{Tors. Then the graded Euler characteristic χ gr pSHpM, γqq " χ gr pSH g pM, γqq P ZrHs{˘H is independent of the choice of the fixed genus g of closures.
Proof. From Corollary 3.42 and Theorem 4.8, we know χ gr pSH g pM, γqq " χ gr pSF HpM, γqq P ZrHs{˘H, where the right hand side is independent of the choice of the fixed genus g of closures.
Proof of Theorem 4.8. First we can attach product 1-handles disjoint from S 1 , . . . , S n . From [BS16a, Section 4.2], attaching a product 1-handle does not change the closure and hence does not make any difference to the multi-grading associated to pS 1 , . . . , S n q. Hence we can assume that γ is connected from now on. From [LY20, Section 3.1], we can pick a disjoint union of properly embedded arcs α " α 1 Y¨¨¨Y α m so that (1) for k " 1, . . . , m, we have Bα k X R`pγq ‰ H and Bα k X R´pγq ‰ H, (2) M zintpN pαqq is a handlebody.
For any k " 1, . . . , g, let β k Ă ζ k be a neighborhood of the intersection point ζ k X γ and let where ζ k,˘Ă R˘pγq. Push the interior of β k into the interior of M to make it a properly embedded arc, which we still call β k . Let Let N " M zintpN pβqq, and let γ N be the disjoint union of γ and a meridian for each component of β. It is explained in [LY20, Section 3.2] that pN, γ N q can be obtained from pM, γq by attaching product 1-handles disjoint from S 1 ,. . . ,S m , so there is a canonical identification SHp´M,´γ, pS 1 , . . . , S n q, pi 1 , . . . , i n qq " SHp´N,´γ N , pS 1 , . . . , S n q, pi 1 , . . . , i n qq Let H " M zintpN pα Y βqq. It is straightforward to check that H is a handlebody. Let Γ µ be the disjoint union of γ and a meridian for each component of α Y β. Let the suture Γ 0 be obtained from Γ µ by performing band sums along ζ k,`a nd ζ k,´f or k " 1, . . . , m. See Figure 15. It is straightforward to check that pN, γ N q can be obtained from pH, Γ 0 q by attaching contact 2-handles along the meridians of all components of α. We prove the theorem in the case when m " 1, while the general case follows from a straightforward induction. If m " 1, then α is connected. Suppose µ is the meridian of α. As explained in Subsection 4.2, attaching a contact 2-handle along µ is the same as performing a 0-surgery along a push-off µ 1 of µ. There is an exact triangle associated to the surgeries along µ 1 that is discussed in [LY20, Section 3.2] (see also [GLW19, Section 3.1]): The map C µ is the map associated to the contact 2-handle attachment as discussed in Subsection 4.2. The suture Γ 1 is obtained from Γ 0 by twisting along p´µq once. for j " 1, . . . , n, let S j,H " S j X H. Since µ is disjoint from S j,H for j " 1, . . . , n, the proof of Proposition 4.6 implies there is a graded version of the exact triangle (4.2):

The canonical mod 2 grading
Throughout this section, we focus on special cases of balanced sutured manifolds obtained from connected closed 3-manifolds and knots in them (c.f. Remark 3.36).
Definition 5.1. Suppose that Y is a closed 3-manifold and z P Y is a basepoint. Let Y p1q be obtained from Y by removing a 3-ball containing z and let δ be a simple closed curve on BY p1q -S 2 . Suppose that K Ă Y is a knot and w is a basepoint on K. Let Y pKq be the knot complement of K and let γ " m Y p´mq consist of two meridians with opposite orientations of K near w. Then pY p1q, δq and pY pKq, γq are balanced sutured manifolds. Define r HpY, zq :" SHpY p1q, δq and KHpY, K, wq :" SHpY pKq, γq.
Convention. Different choices of the basepoints give isomorphism vector spaces. Since we only care about the isomorphism class of the vector spaces, we omit the basepoints and simply write r HpY q and KHpY, Kq instead.
To be more specific and consistent with [LY20], in this section, we focus on instanton theory. Based on the discussion in Subsection 2.1, we specify the Floer homology HpY q and the cobordism map HpW q to be I ω pY q and IpW, νq. For a connected closed 3-manifold, the framed instanton Floer homology I 7 pY q defined in [KM11a] is isomorphic to r HpY q when H is instanton theory. Hence we replace r HpY q by I 7 pY q throughout this section. Also we replace SH and KH by SHI and KHI, respectively. Recall that the definitions of SHI and KHI a priori depend on the choice of a fixed and large genus g of closures. We write SHI g and KHI g explicitly in this section. However, for instanton theory, closures of different genus induce isomorphic groups and we can use closures of genus one to define sutured instanton homology (c.f. [KM10b,Section 7]).
In this section, we discuss the canonical Z 2 -grading on KHI g and the decomposition of I 7 in Theorem 1.18. For other Floer-type theory, the construction in [LY20, Section 4.3] can be adapted without essential changes and we have a decomposition for r HpY q similar to that in Theorem 1.18. The results in this section also apply without essential changes except arguments about different genera of closures, which use the Floer's excision theorem along a surface of genus one.
5.1. The case of an unknot.
In this subsection, we study the model case: the unknot U in S 3 . Suppose µ U and λ U are the meridian and the longitude of U , respectively. The knot complement is identified with a solid torus S 1ˆD2 : where ρpµ U q " S 1ˆt 1u, and ρpλ U q " t1uˆBD 2 . For co-prime integers x and y, let γ px,yq " γ xλ U`y µ U Ă BS 3 pU q be the suture consists of two disjoint simple closed curves representing˘pxλ U`y µ U q.
Convention. Note γ px,yq " γ p´x,´yq . From term (4) in Proposition 2.30, the orientation of the suture does not influence the isomorphism type of formal sutured homology. Hence we do not care about the orientation of the suture and we always assume y ě 0.
We describe a closure of the balanced sutured manifold pS 3 pU q, γ px,yq q as follows. Let Σ be a connected closed surface of genus g ě 1. Suppose Y Σ " S 1ˆΣ and Σ " t1uˆΣ. Pick a non-separating simple closed curve α Ă Σ and suppose its complement is Y Σ pαq " Y Σ zintpN pαqq. There is a framing on BY Σ pαq induced by the surface Σ. Let µ α and λ α be the corresponding meridian and longitude, respectively. Also, suppose p P Σ is a point disjoint from α. According to the discussion in Section 2, we can form a closure pȲ , R, ωq of pS 3 pU q, γ px,yq q as follows: (5.2)Ȳ " S 3 pU q Y φ Y Σ pαq, R " Σ, and ω " S 1ˆt ptu, where φ : BS 3 pU q -Ý Ñ BY pαq is an orientation reversing diffeomorphism such that Note that different choices of the preimage of µ α lead to different closures of pS 3 pU q, γ px,yq q. From (5.3), we know that φpλ U q " zλ α`y µ α , where z " x if y " 0, z " y is arbitrary if y " 1, and zx " 1 pmod yq in other cases. Again different choices of z lead to different closures. From now on, we fix the value of z as follows: z " x of y " 0, z " 0 if y " 1, and z is the minimal positive integer so that y|pxz´1q. Now, composing φ with the inverse of the map ρ in (5.1), suppose where φ˝ρ´1 : BpS 1ˆD2 q Ñ BY pαq is a diffeomorphism such that φ˝ρ´1pt1uˆBD 2 q " zλ α`y µ α .
Hence,Ȳ is obtained from Y by performing a y{z surgery and we also writeȲ " p Y y{z .
Proof. First, we can focus on the closure pȲ " p Y y{z , R " Σ, ωq as in (5.2). We need to compute the Euler characteristic of SHI g pS 3 pU q, γ px,yq q :" I ω p p Y y{z |Σq.
If y " 1, then z " 0 and p Y 1{0 " S 1ˆΣ . By Axiom (A1-5), we have If y ą 1, we have y ą z ě 1. If z " 1, then we have an exact triangle from Axiom (A2) where the parity of the map f is odd and those of the rest two are even by Proposition 2.4. Hence we conclude by induction that χpI ω p p Y y |Σqq "´y. Finally, when y ą z ą 1, suppose the continued fraction of´y{z iś y z " ra 0 , . . . , a n s " a 0´1 a 1´1¨¨´1 an , where a n ď´2. Define´y 1 z 1 " ra 0 , . . . , a n´1 s and´y 2 z 2 " ra 0 , . . . , a n´1`1 s, where y 1 , y 2 ě 0. From a basic property of continued fraction, we have y " y 1`y2 and z " z 1`z2 .
From Axiom (A2), there exists an exact triangle where the parity of the map f is odd and those of the rest two are even by Proposition 2.4.
Remark 5.3. It is worth mentioning that different papers have different normalization for the canonical Z 2 -grading. Our choice of normalization in Axiom (A3) is the same as in [KM10a]. In Lidman, Pinzón-Caicedo, and Scaduto's setup [LPCS20], they adapted another normalization and proved χpI ω pS 1ˆΣ |Σqq " 1 for Σ of any genus that is at least one.
Proof. This corollary follows directly from the fact that canonical maps from I ω p p Y 1 |Σq to I ω 1 pY 1 |R 1 q is a composition of cobordism maps and hence is homogeneous.

Sutured knot complements.
Suppose Y is a closed 3-manifold and K Ă Y is a null-homologous knot. Any Seifert surface S of K gives rise to a framing on BY pKq: the meridian µ can be picked as the meridian of the solid torus N pKq, and the longitude λ can be picked as S X BY pKq. The 'half lives and half dies' fact for 3-manifolds implies that the following map has a 1-dimensional image: B˚: H 2 pY pKq, BY pKq; Qq Ñ H 1 pBY pKq; Qq.
Hence any two Seifert surfaces lead to the same framing on BY pKq.
Definition 5.5. The framing pλ,´µq defined as above is called the canonical framing of pY, Kq. With this canonical framing, let γ px,yq " γ xλ`yµ Ă BY pKq be the suture consists of two disjoint simple closed curves representing˘pxλ`yµq.
Our goal in this subsection is to define a canonical Z 2 -grading on SHI g pY pKq, γ px,yq q for any fixed large enough g. Recall SHI g pM, γq is the projective transitive system formed by closures of pM, γq of a fixed genus g. We first assign a Z 2 -grading for any closure of pY pKq, γ px,yq q.
Suppose pȲ , R, ωq is a closure of pY pKq, γ px,yq q. Then we can form a closure pȲ U , R, ωq of pS 3 pU q, γ px,yq q by taking Here id is the diffeomorphism between toroidal boundaries which respect the canonical framings on both boundaries.
Definition 5.6. The modified Z 2 -grading on I ω pȲ |Rq is defined as follows.
(1) If χpI ω pȲ U |Rqq is negative, then the grading is defined by the canonical Z 2 -grading on I ω pȲ |Rq.
(2) If χpI ω pȲ U |Rqq is positive, then the grading is defined by switching the odd and even parts of I ω pȲ |Rq with the canonical Z 2 -grading.
Suppose pȲ , R, ωq and pȲ 1 , R, ωq are two closures of pY pKq, γ px,yq q so thatȲ 1 is obtained fromȲ by a Dehn surgery along a curve β ĂȲ , which is disjoint from intpM q, R and ω. Then there is a map F : I ω pȲ |Rq Ñ I ω pȲ 1 |Rq associated to the Dehn surgery along β ĂȲ . Let pȲ U , R, ωq and pȲ 1 U , R, ωq be the closures of pS 3 pU q, γ px,yq q constructed as in (5.4). There is also a map F U : I ω pȲ U |Rq Ñ I ω pȲ 1 U |Rq associated to the same Dehn surgery along β ĂȲ U . Then we have the following.
Lemma 5.7. The maps F and F U have the same parity with respect to the canonical Z 2 -gradings on corresponding instanton Floer homologies.
Proof. Note that H 1 pS 3 pU q; Qq -Qxµ U y and the map i Ů : H 1 pBS 3 pU q; Qq Ñ H 1 pS 3 pU q; Qq induced by the inclusion has a 1-dimensional kernel generated by λ U . For a null-homologous knot K Ă Y , we know that the map i˚: H 1 pBY pKq; Qq Ñ H 1 pY pKq; Qq induced by the inclusion has a 1-dimensional kernel generated by the longitude λ of K and has a 1-dimensional image generated by the meridian µ of K. Hence, from the Mayer-Vietoris sequence, we know that there is an injective map j : H 1 pȲ U ; Qq ãÑ H 1 pȲ ; Qq, that sends rµ U s to rµs and sends every homology class inȲ U zS 3 pU q "Ȳ zY pKq using the natural map i 1 : H 1 pȲ zY pKq; Qq Ñ H 1 pȲ ; Qq.
Similarly, since β X intpY pKqq " H, we know that there is an injective map kerpι Ů q " kerpι˚q.
Since F and F U are associated to Dehn surgeries along β of the same slopes, we conclude from 2.4 that their parity must be the same.
Proof. From the definition of Φ and Axiom (A3-3), we know that Φ is always homogeneous. For two closures pȲ , R, ωq and pȲ 1 , R 1 , ω 1 q of pY pKq, γ px,yq q, we can form two corresponding closures pȲ U , R, ωq and pȲ 1 U , R 1 , ω 1 q of pS 3 pU q, γ px,yq q as in (5.2), respectively. There is a canonical map Φ U : I ω pȲ U |Rq -Ý Ñ I ω 1 pȲ 1 U |R 1 q. From Definition 5.6, the modified Z 2 -gradings on I ω pȲ |Rq and I ω pȲ 1 |R 1 q coincide if and only if the canonical Z 2 -gradings on I ω pȲ U |Rq and I ω pȲ 1 U |R 1 q coincide. The definition of the modified Z 2 -grading automatically makes Φ U even under the modified Z 2 -grading. Hence to show that Φ is grading preserving, it suffices to show that Φ and Φ U have the same parity under the modified Z 2 -grading.
From the construction of the canonical maps, there is a sequence of simple closed curves β 1 , . . . , β n on R, such that the map Φ is the composition of cobordism maps induced by a diffeomorphism and the sequence of Dehn surgeries. Similarly, the map Φ U is the composition of the maps induced another diffeomorphism and the same sequence of Dehn surgeries onȲ U . Since the surgery curves are all on R and disjoint from intpY pKqq, cobordism maps induced by diffeomorphisms are always with even degrees, i.e. preserving the Z 2 -grading. For Dehn surgeries along β i , we can apply Lemma 5.7 and then finish the proof.
Definition 5.9. Suppose pY pKq, γ px,yq q is the balanced sutured manifold constructed as before and suppose g is the fixed large enough genus of closures. By Lemma 5.8, we can define the canonical Z 2 -grading on SHI g pY pKq, γ px,yq q by the modified Z 2 -grading on the closures. In particular, there is a canonical Z 2 -grading on KHI g pY, Kq.
Then the bypass exact triangle in Theorem 2.19 becomes the following.
Proposition 5.14. Suppose K Ă Y is a null-homologous knot, and suppose the surgery slopes y i {x i for i " 1, 2, 3 are defined as in Definition 5.12. Suppose ψ˚,˚and ψ˚,˚are from two different bypasses, where˚means the corresponding slope. Then there are two exact triangles about ψ˚,˚and ψ˚,˚, respectively. As stated in Proposition 2.20, the bypass maps ψ 1 , ψ 2 , and ψ 3 are induced by some cobordism maps. Then we have the following.
Lemma 5.15. Suppose g is a large enough integer and suppose y i {x i for i " 1, 2, 3 is from Definition 5.12. Suppose further x 1 " x 2`x3 and y 1 " y 2`y3 .
With respect to the canonical Z 2 -grading on SHI g in Definition 5.9, the parity of the map ψ 2 is odd and those of the rest two are even. As a consequence, χpSHI g p´Y pKq,´γ px1,y1q qq " χpSHI g p´Y pKq,´γ px3,y3q qq`χpSHI g p´Y pKq,´γ px2,y2q qq.
Proof. As in Proposition 2.20, we can fix a large enough g so that for i " 1, 2, 3, there are closures pȲ i , R, ωq for p´Y pKq,´γ pxi,yiq q of genus g, and the bypass maps ψ 1 , ψ 2 , and ψ 3 have the same Z 2 degree because the maps induced by Dehn surgeries along three curves ζ 1 , ζ 2 , and ζ 3 in corresponding closuresȲ i zintpY pKqq. Since we only care about the Z 2 degrees of maps, in a slight abuse of notation, we do not distinguish the bypass map and the map induced by Dehn surgery. For i " 1, 2, 3, we can form corresponding closures pȲ U i , R, ωq as in (5.2) so that the curves ζ 1 , ζ 2 , and ζ 3 still lie in closures. Moreover, suitable surgeries along these curves induces an exact triangle SHI g p´S 3 pU q,´γ px3,y3q q ψ U 3 / / SHI g p´S 3 pU q,´γ px1,y1q q ψ U 1 t t SHI g p´S 3 pU q,´γ px2,y2q q ψ U 2 j j As in the proof of Lemma 5.8, with the help of Lemma 5.7, it suffices to check that the parities of maps ψ U 1 , ψ U 2 , and ψ U 3 are odd or even as claimed, with respect to the canonical Z 2 -grading on SHI g from Definition 5.9.
Note that the maps ψ U i for i " 1, 2, 3 are coming from a real surgery exact triangle as in Proposition 2.20, while the Z 2 -gradings on SHI g could possibly be shifted due to the normalization in Definition 5.6 and the surgery along curves η 1 and η 2 as in Proposition 2.20. Hence they still satisfies the hypothesis of Lemma 2.5. Thus, we conclude that the parity of ψ U 2 is odd and those of the other two are even. Similarly, the parity of ψ 2 is odd and those of the other two are even, and we have χpSHI g p´Y pKq,´γ px1,y1q qq " χpSHI g p´Y pKq,´γ px3,y3q qq`χpSHI g p´Y pKq,´γ px2,y2q qq.
Let Y be a closed 3-manifold and let K Ă Y be a null-homologous knot. Suppose S is a minimal genus Seifert surface of K. Its genus is always denoted by gpSq, which is distinguished with g, the fixed genus of closures. We refer [LY20, Section 4] for the definitions of sutures Γ n , Γ n py{xq, the admissible surface with stablization S τ , the bypass maps ψ˚,˚, ψ˚,˚, and numbers im ax , im in . To simplify our notation, we write (5.5) χ g y{x p´Y, K, iq " χpSHI g p´Y pKq,´γ px,yq , S τ , iqq, where the Euler characteristic is with respect to the canonical Z 2 -grading on SHI g as in Definition 5.9. We write (5.6) χ g y{x p´Y, Kq " ÿ iPZ χ g y{x p´Y, K, iq When |x| " 1, we write y{x as an integer. Also, we write χ g µ p´Y, K, iq " χ g 1{0 p´Y, K, iq to specify the meridional suture.
Lemma 5.16. Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot. For g P Z large enough and any i P Z, we have χ g 1 p´Y, K, iq " χ g µ p´Y, K, iq and χ g 0 p´Y, K, iq " 0.
Hence we conclude by (5.5) and (5.6) that χ g 1 p´Y, K, gpSqq " χ g µ p´Y, K, gpSqq and χ g 0 p´Y, K, gpSqq " 0. The lemma follows from the induction on the grading i.
Lemma 5.17. Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot. For the suture γ px,yq with y ą 0, we know that χ g y{x p´Y, K, iq " y´1 ÿ j"0 χ g µ p´Y, K, i´i y max`i µ max`j q.
Proof. We only prove the case when x ă 0. The other case is similar. First, if x " 1, then we have a bypass exact triangle (in this case we write y " n) SHI g p´Y pKq,´Γ n , S τ , iq ψ ń ,n`1 / / SHI g p´Y pKq,´Γ n`1 , S τ , iq ψ n`1 ,µ r r SHI g p´Y pKq,´Γ µ , S τ , i`1q If x ą 1, we can use the continued fraction description of y{x and apply an induction in the same spirit as in the proof of Lemma 5.2.
Corollary 5.18. Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot. For the suture γ px,yq where y ą 2gpSq, and for any i P Z so that i y max´2 gpSq ě i ě i y min`2 gpSq, we know that χ g y{x p´Y, K, iq " χ g µ p´Y, Kq.
Proof. The corollary follows immediately from Lemma 5.17 and the fact that there are only p2gpSq`1q gradings with nontrivial elements for SHI g p´Y pKq,´Γ µ , S τ q.
Lemma 5.19. Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot. Then we have χ g µ p´Y, Kq "˘χpI 7 p´Y qq. Proof. From Lemma 5.17, we know that for any n P Z ě0 , we have χ g n p´Y, Kq " n¨χ g µ p´Y, Kq. From [LY20, Lemma 4.11], we know that there is an exact triangle SHI g p´Y pKq,´Γ n q / / SHI g p´Y pKq,´Γ n`1 q v v I 7 p´Y q h h Hence, by Lemma 2.5, we know that there is a proper sign assignment for all n so that χ g n p´Y, Kq˘χ g n`1 p´Y, Kq˘χpI 7 pY qq " 0. Hence the only possibilities are χpI 7 pY qq "˘χ g µ p´Y, Kq.
Proof of Proposition 1.19. It is an immediate corollary following Corollary 5.18, Lemma 5.19 and the definition of the decomposition from [LY20, Section 4.3].
For a knot K in S 3 , we can actually fix the sign ambiguity coming from different choices of the fixed genus of the closures.
Note that this coincide with our choice of modified Z 2 -grading: when the Euler characteristic is negative, we do not perform any shift.
For general g, it is straightforward to generalize the above construction forȲ 1 andȲ 2 toȲ g and Y g`1 . There is a similar cobordism W g fromȲ g Y S 1ˆΣ 2 toȲ g`1 , the degree of which can be computed easily to be odd. Hence by induction we conclude that for all g, χ g µ p´S 3 , Kq "´1.
By Lemma 5.20, we can identify χ g µ p´S 3 , Kq for all large enough g, we simply write χ µ p´S 3 , Kq " χ g µ p´S 3 , Kq instead. Applying Lemma 5.17, we know that for any g large enough and y ą 0, χ g y{x p´S 3 , Kq "´y. Similarly, we simply write χ y{x p´S 3 , Kq instead.
Finally, we consider the projectively system SHIpM, γq for a balanced sutured manifold pM, γq defined in [BS15], which is independent of the choice of the genus of the closures. The isomorphism class of SHIpM, γq and SHI g pM, γq are the same. Similar to SHI g pM, γq, it has a decomposition associated to an admissible surface S Ă pM, γq.
Definition 5.21. Suppose pM, γq is a balanced sutured manifold and S is an admissible surface in pM, γq. For any i, j P Z, define SHIpM, γ, S, iqrjs " SHIpM, γ, S, i´jq.
In [Li19, Section 5], the first author constructed a minus version of the instanton knot homology via the direct system (5.7)¨¨Ñ SHIp´S 3 pKq, Γ n , S τ qrgpKq´i n max s ψ ń ,n`1 Ý ÝÝÝÝ Ñ SHIp´S 3 pKq, Γ n`1 , S τ qrgpKq´i n`1 max s Ñ¨¨ä nd define KHI´p´S 3 , Kq to be the direct limit of (5.7). All ψ ń ,n`1 are grading preserving after shifting, so there is a well-defined Z-grading (the Alexander grading) on KHI´p´S 3 , Kq, which we write as KHI´p´S 3 , K, iq. SHIp´S 3 pKq, Γ n`1 , S τ qrgpKq´i n`1 max s ψ n`1 ,n`2 SHIp´S 3 pKq, Γ n`1 , S τ qrgpKq´i n`1 max s ψ n`1 ,n`2 / / SHIp´S 3 pKq, Γ n`2 , S τ qrgpKq´i n`2 max s Hence the maps tψ ǹ ,n`1 u induces an map U on KHI´p´S 3 , Kq. Proof of Proposition 1.21. From Lemma 5.15, the parity of maps ψ ń ,n`1 are all even, hence there is a well-defined Z 2 -grading on KHI´p´S 3 , Kq. Again from Lemma 5.15, we know that the parity of the map U is even, i.e., preserving the Z 2 -grading on KHI´p´S 3 , Kq. Finally, we can apply Lemma 5.17 and the fact χpKHI g p´S 3 , Kqq "´∆ K ptq to conclude the desired formula. Note that by our the normalization, the sign is negative.