Bennequin-Plamenevskaya-Shumakovitch type inequalities for Kronheimer-Mrowka's concordance invariant

TL;DR

This paper establishes new lower bounds for Kronheimer-Mrowka's concordance invariant using techniques inspired by classical inequalities and computations for torus knots.

Contribution

It introduces Bennequin-Plamenevskaya-Shumakovitch type inequalities for the invariant $s^{\#}$, expanding the understanding of its properties in knot concordance.

Findings

Derived lower bounds for $s^{\#}$ using torus knot computations.

Connected cobordism inequalities with classical inequalities for slice-torus invariants.

Extended the framework for analyzing the $s^{\#}$ invariant in knot theory.

Abstract

We give Bennequin-Plamenevskaya-Shumakovich type lower bounds for the concordance invariant $s^{\#}$ introduced by Kronheimer and Mrowka. The proof is a consequence of computations for torus knots and the cobordism inequality of $s^{\#}$ due to Gong, combined with well-known arguments used for slice-torus invariants.