On the tau invariants in instanton and monopole Floer theories

TL;DR

This paper unifies two approaches to tau invariants in instanton and monopole Floer theories, establishing their equivalence, exploring implications for Heegaard Floer theory, and computing invariants for twist knots.

Contribution

It identifies two definitions of tau invariants in Floer theories, showing they are equivalent, and applies this to compute invariants for specific knots.

Findings

Proves the equivalence of two tau invariant definitions.

Establishes a relationship with Heegaard Floer theory.

Computes tau invariants for twist knots.

Abstract

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying $\tau_{\mathrm{G}}$, defined by the second author via the minus flavors $\underline{\operatorname{KHI}}^-$ and $\underline{\operatorname{KHM}}^-$ of the knot homologies, with $\tau_{\mathrm{G}}^{\sharp}$, defined by Baldwin and Sivek via cobordism maps of the $3$-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute $\underline{\operatorname{KHI}}^-$ and $\underline{\operatorname{KHM}}^-$ for twist knots.