We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for a positive rational number and a nontrivial knot in the -sphere, there exists an irreducible homomorphism
unless and is both fibered and strongly quasipositive, broadly generalizing results of Kronheimer and Mrowka. We also answer a question of theirs from 2004, proving that there is always an irreducible homomorphism from the fundamental group of 4-surgery on a nontrivial knot to . In another application, we show that a slight enhancement of the -polynomial detects infinitely many torus knots, including the trefoil.
Cite this article
John A. Baldwin, Steven Sivek, Instantons and L-space surgeries. J. Eur. Math. Soc. 25 (2023), no. 10, pp. 4033–4122DOI 10.4171/JEMS/1280