On the slice-torus invariant $q_M$ from $\mathbb{Z}_2$-equivariant Seiberg--Witten Floer cohomology

TL;DR

This paper demonstrates that the slice-torus invariant $q_M$ derived from $ ext{Z}_2$-equivariant Seiberg--Witten Floer cohomology cannot be expressed as a linear combination of several well-known knot invariants, highlighting its distinct nature.

Contribution

It establishes the independence of the $q_M$ invariant from other prominent knot invariants, showing it captures unique information in knot concordance.

Findings

$q_M$ cannot be written as a linear combination of other invariants.

Shows the uniqueness of $q_M$ in the landscape of knot invariants.

Highlights the limitations of existing invariants in capturing $q_M$'s information.

Abstract

We show that Iida--Taniguchi's $\mathbb{Z}$-valued slice-torus invariant $q_M$ cannot be realized as a linear combination of Rasmussen's $s$-invariant, Ozsv\'ath--Szab\'o's $\tau$-invariant, all of the $\mathfrak{sl}_N$-concordance invariants ($N \geq 2$), Baldwin--Sivek's instanton $\tau$-invariant, Daemi--Imori--Sato--Scaduto--Taniguchi's instanton $\tilde{s}$-invariant and Sano--Sato's Rasmussen type invariants $\tilde{ss}_c$.