Framed instanton homology and Frøyshov's invariant

TL;DR

The paper determines the framed instanton homology $I^\#(S^3_{p/q}(K);\mathbb F)$ for Dehn surgeries on knots in $S^3$ with $\mathbb F=\mathbb Z/2$, and relates its dimension to Frøyshov's $q_3$ invariant. It develops a robust algebraic–topological framework built from the tilde complex, equivariant instanton homology, and surgery exact triangles to extend integer-slope results to all rational slopes, introducing invariants $M(K)$ and $r_2(K)$ and proving additivity under connected sum. A key outcome is that the dimension graph of $I^\#(S^3_r(K);\mathbb F)$ is always '$W$-shaped$,$ with $M(K)$ controlling the transition values and $q_3$ encoding obstruction data through odd slopes. The authors also define and relate torsion-averse knots to instanton $L$-space knots, derive lower/upper bounds for $I^\#(S^3_r(K))$ across slopes, and provide detailed computations for notable knots, yielding new obstructions to nondegenerate $SU(2)$-abelian surgeries. Collectively, the results deepen the connection between framed instanton homology, Frøyshov-type invariants, and knot concordance data, with concrete applications to $SU(2)$-abelian surgery problems.

Abstract

We determine the framed instanton homology with coefficients in $\mathbb F = \mathbb Z/2$ for Dehn surgeries on a knot in the $3$-sphere. The dimension of these groups is seen to have a close relationship with a homology cobordism invariant due to Froyshov. As an application, we show that $r$-surgery on a non-trivial knot cannot be nondegenerate $SU(2)$-abelian for any $|r| \le 4\lceil g(K)/2\rceil$, which is $2g(K)$ for $g$ even and $2g(K) + 2$ for $g$ odd.

Theorems & Definitions (79)

• Conjecture 1.1
• Theorem 1.2: bs-concordance*Theorem 1.1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• proof
• Corollary 1.7
• proof
• Corollary 1.8