[Paper Review] Instantons and L-space surgeries

John A. Baldwin, Steven Sivek|arXiv (Cornell University)|Oct 29, 2019

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes that instanton L-space knots are fibered and strongly quasipositive using a novel decomposition theorem for cobordism maps in framed instanton Floer homology. It proves that for nontrivial knots in S³, Dehn surgery with rational slope r ≥ 2g(K)−1 is SU(2)-abelian only if K is fibered and strongly quasipositive, broadly generalizing Kronheimer and Mrowka's Property P result and confirming the existence of irreducible SU(2) representations for 4-surgery on nontrivial knots.

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 $ extrm{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 \[π_1(S^3_r(K)) o SU(2)\] 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.

Motivation & Objective

• To establish that instanton L-space knots are fibered and strongly quasipositive using gauge-theoretic techniques in instanton Floer homology.
• To generalize Kronheimer and Mrowka's result on SU(2)-abelian surgeries beyond r ≤ 2, particularly for r ≥ 3.
• To resolve a 2004 question of Kronheimer and Mrowka by proving that 4-surgery on any nontrivial knot admits an irreducible SU(2) representation.
• To show that a slight enhancement of the A-polynomial detects infinitely many torus knots, including the trefoil.
• To provide strong restrictions on SU(2)-averse knots and their limit slopes, showing they must be fibered and strongly quasipositive with limit slope > 2g(K)−1.

Proposed method

• Introduces a new decomposition theorem for cobordism maps in framed instanton Floer homology, analogous to Spin^c decompositions in other Floer theories.
• Applies this decomposition to study the instanton Floer homology of surgeries on knots, particularly focusing on the odd-dimensional instanton homology of the 0-surgery manifold.
• Uses the Euler characteristic and eigenspace dimensions of the action of a surface class on the instanton homology to constrain the possible knot types.
• Employs results from Klassen and Lin on SU(2) representations and character varieties to analyze the structure of the representation variety R(Y) for Dehn surgeries.
• Applies the theory of thin knots and their genus bounds to rule out non-torus knots under certain r-thinness conditions.
• Leverages the fact that if a knot is r-thin and satisfies certain number-theoretic conditions on r, then it must be a torus knot, using genus additivity in satellite constructions.

Experimental results

Research questions

• RQ1 Under what conditions on the rational surgery slope r is the 3-manifold S³_r(K) not SU(2)-abelian for a nontrivial knot K in S³?
• RQ2 Can the existence of irreducible SU(2) representations in Dehn surgeries be established for r = 3 and r = 4, particularly when r ≥ 2g(K)−1?
• RQ3 Do fibered, strongly quasipositive knots with g(K) = 2 and r = 3 yield SU(2)-abelian surgeries, and if so, are there any such knots beyond the trefoil?
• RQ4 Can the A-polynomial or its enhancement detect specific torus knots, such as T_{p,q} for prime p, q?
• RQ5 What constraints does the SU(2)-representation variety impose on SU(2)-averse knots, particularly regarding their limit slopes and topological type?

Key findings

• Instanton L-space knots are proven to be fibered and strongly quasipositive, providing a new characterization via instanton Floer homology.
• For any nontrivial knot K, S³_4(K) admits an irreducible SU(2) representation, confirming a conjecture of Kronheimer and Mrowka.
• S³_3(K) is not SU(2)-abelian unless K is fibered, strongly quasipositive, and has genus 2.
• The enhanced A-polynomial detects infinitely many torus knots, including T_{p,q}, T_{p²,q}, T_{p²,q²}, T_{4,q} (q > 3), and T_{4,q²}.
• Any SU(2)-averse knot must be fibered, and either the knot or its mirror is strongly quasipositive with limit slope strictly greater than 2g(K)−1.
• If r > 12 is square-free and odd with at least two distinct prime divisors, or r = p²q or p²q² for distinct primes p, q with q ≥ 3, then any r-thin knot must be a torus knot.