Instantons and L-space surgeries

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 ${\text{Spin}}^c$ 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 \[π_1(S^3_r(K)) \to SU(2)\] unless $r\ge 2g(K)-1$ 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.

arXiv comment79 pages, 1 figure; v2: accepted version

References in corpus (11)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Knot Floer homology detects fibred knots
• Corrigendum to "Knot Floer homology detects fibred knots"
• Knot homology groups from instantons
• Non-triviality of the A-polynomial for knots in S^3
• Instantons and odd Khovanov homology
• Dehn surgery, the fundamental group and SU(2)
• Detection of knots and a cabling formula for A-polynomials
• Some remarks on cabling, contact structures, and complex curves
• SU(2)-cyclic surgeries and the pillowcase
• Addendum to: "Knots, sutures and excision"

Cited by in corpus (19)

• Framed instanton homology and concordance
• Instanton Floer homology, sutures, and Heegaard diagrams
• SU(2)-cyclic surgeries and the pillowcase
• An enhanced Euler characteristic of sutured instanton homology
• Instanton Floer homology of almost-rational plumbings
• Small Dehn surgery and SU(2)
• Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil
• L-space knots are fibered and strongly quasipositive
• SU(2) representations and a large surgery formula
• A unified Casson-Lin invariant for the real forms of SL(2)
• Knot surgery formulae for instanton Floer homology II: applications
• Framed instanton homology and concordance, II
• 2-torsion in instanton Floer homology
• Torus knots, the A-polynomial, and SL(2,C)
• Small Heegaard genus and SU(2)
• Chainmail links and L-spaces
• An instanton take on some knot detection results
• Knot surgery formulae for instanton Floer homology I: the main theorem
• The SO(3) Vortex Equations over Orbifold Riemann Surfaces