Equivariant concordance invariance of s_tau

Establish that the numerical invariant s_tau, defined from the spectral sequence in Theorem 1 for a strongly invertible knot (K, τ) as the Maslov grading of the Alexander-grading-0 generator that survives to the E_∞ page, is invariant under equivariant concordance of directed strongly invertible knots (i.e., s_tau is unchanged for knots related by an equivariant concordance as defined in the paper).

Background

The paper proves a localization spectral sequence (Theorem 1) from the knot Floer homology of a strongly invertible knot K to the Floer homology of S3. From this spectral sequence the author defines a numerical invariant s_tau by taking the Maslov grading of the surviving Alexander-grading-0 generator at E_∞.

The authors note parallels with Rasmussen’s s and other concordance invariants derived from Floer theories, and propose s_tau as a candidate for an equivariant concordance invariant in the strongly invertible setting.

References

From Theorem \ref{Mainthm1} we extract a numerical invariant $s_\tau$ of strongly invertible knots constructed similarly to many concordance invariants including Rasmussen's $s$ invariant , and Hendricks-Lipshitz-Sarkar's $q_\tau(K)$ and $d_\tau(K,m)$ , and conjecture that this invariant is an equivariant concordance invariant.

— Localization and the Floer homology of strongly invertible knots (2408.13892 - Parikh, 25 Aug 2024) in Section 1.1 (Main results)