Instantons and L-space surgeries
John A. Baldwin, Steven Sivek
Abstract
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 \operatorname{Spin}^c decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for r a positive rational number and K a nontrivial knot in the 3 -sphere, there exists an irreducible homomorphism \pi_1(S^3_r(K)) \to SU(2) unless r \geq 2g(K)-1 and K 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 SU(2) . In another application, we show that a slight enhancement of the A -polynomial detects infinitely many torus knots, including the trefoil.