Instanton 2-torsion and fibered knots

TL;DR

This paper proves that the unreduced singular instanton homology of fibered knots generally contains 2-torsion, introduces a sutured instanton formula, and explores implications for knot invariants and relations to Heegaard Floer theory.

Contribution

It establishes the presence of 2-torsion in instanton homology for fibered knots and provides a sutured instanton formula to compare homology dimensions, advancing understanding of knot invariants.

Findings

$I^lat(Y,K;\mathbb{Z})$ has 2-torsion for most fibered knots.

$I^lat(S^3,K;\mathbb{C})$ is determined by the Alexander polynomial for knots with lens space surgeries.

The next-to-top Alexander grading of instanton knot homology is non-vanishing for knots with unknotting number one.

Abstract

We prove that the unreduced singular instanton homology $I^\sharp(Y,K;\mathbb{Z})$ has $2$-torsion for any null-homologous fibered knot $K$ of genus $g>0$ in a closed $3$-manifold $Y$ except for $\#^{2g}S^1\times S^2$. The main technical result is a formula of $I^\sharp(Y,K;\mathbb{C})$ via sutured instanton theory, by which we can compare the dimensions of $I^\sharp(Y,K;\mathbb{F}_2)$ and $I^\sharp(Y,K;\mathbb{C})$. As a byproduct, we show that $I^\sharp(S^3,K;\mathbb{C})$ for a knot $K\subset S^3$ admitting lens space surgeries is determined by the Alexander polynomial, while some special cases of torus knots have been previously studied by many people. Another byproduct is that the next-to-top Alexander grading summand of instanton knot homology $KHI(S^3,K,g(K)-1)$ is non-vanishing when $K$ has unknotting number one, which generalizes the Baldwin--Sivek's result in the fibered case. Finally, we discuss the relation to the Heegaard Floer theory.