Instanton 2-torsion and Dehn surgeries

TL;DR

This paper extends the understanding of 2-torsion in instanton Floer homology to rational surgeries, establishing new obstructions for knots based on their instanton homology and small-surgery properties.

Contribution

It generalizes previous results from integral to rational surgeries and introduces new small-surgery obstructions related to $SU(2)$-abelian properties.

Findings

If $I^{lat}(S^3_r(K))$ is 2-torsion-free, then $K$ is an instanton L-space knot and $r>2g(K)-1.

Certain small surgeries imply $K$ is the unknot or trefoil, sharpening existing theorems.

Extension of 2-torsion analysis to all rational surgeries in instanton Floer homology.

Abstract

In our earlier work on $2$-torsion in instanton Floer homology, we considered only integral surgeries on a knot $K\subset S^3$ and showed that the absence of $2$-torsion forces $K$ to be fibered. The present paper extends the result to all rational surgeries. We prove that if the framed instanton homology $I^{\sharp}(S^3_r(K);\mathbb{Z})$ is $2$-torsion-free for some $r\in \mathbb{Q}_+$, then $K$ is an instanton L-space knot and $r>2g(K)-1$. Leveraging this $2$-torsion perspective, we also obtain new small-surgery obstructions: If either $S^{3}_{5}(K)$ or $S^{3}_{11/2}(K)$ is $SU(2)$-abelian, then $K$ must be the unknot or the right-handed trefoil. This result sharpens the small-$SU(2)$-abelian surgery theorems of Kronheimer--Mrowka, Baldwin--Sivek, and Baldwin--Li--Sivek--Ye.