Framed instanton homology and Fr{\o}yshov's invariant

TL;DR

This paper calculates the framed instanton homology for Dehn surgeries on knots in the 3-sphere, revealing its connection to Froyshov's invariant and applying it to constraints on $SU(2)$-abelian surgeries.

Contribution

It explicitly determines the framed instanton homology for Dehn surgeries and links it to a homology cobordism invariant, providing new bounds on $SU(2)$-abelian surgeries.

Findings

Dimension of instanton homology relates to Froyshov's invariant.

Certain surgeries cannot be nondegenerate $SU(2)$-abelian.

Bounds depend on the genus of the knot.

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.