Non ambiguous structures on 3-manifolds and quantum symmetry defects

Abstract

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped 3-manifolds can be split in a “symmetrization” factor and a “reduced” state sum. We show that these factors are invariants on their own, that we call “symmetry defects” and “reduced QHI,” provided the manifolds are endowed with an additional “non ambiguous structure,” a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact 3-manifolds endowed with PSL${}_2(\mathbb{C}$-characters, beyond the case of cuspedmanifolds. Given a manifold $M$ with non empty boundary, we provide a partial “holographic” description of the nonambiguous structures in terms of the intrinsic geometric topology of $\partial M$. Special instances of non ambiguous structures can be defined by means of taut triangulations, and the symmetry defects have a particularly nice behaviour on such “taut structures.” Natural examples of taut structures are carried by any mapping torus with punctured fibre of negative Euler characteristic, or by sutured manifold hierarchies. For a cusped hyperbolic 3-manifold $M$ which bres over $S^1$, we address the question of determining whether the fibrations over a same fibered face of the Thurston ball define the same taut structure. We describe a few examples in detail. In particular, they show that the symmetry defects or the reduced QHI can distinguish taut structures associated to different fibrations of $M$. To support the guess that all this is an instance of a general behaviour of state sum invariants of 3-manifolds based on some theory of 6$j$-symbols, finally we describe similar results about reduced Turaev–Viro invariants.