# On the Kronheimer-Mrowka concordance invariant

## TL;DR

This paper computes the Kronheimer-Mrowka invariant $s^lat$ for various knots, reveals its non-additivity and differences from Rasmussen's $s$, and introduces new related invariants and their properties.

## Contribution

It introduces a detailed computation of $s^lat$ for specific knots, shows its non-additivity, and defines new invariants $s^lat_\pm$ and $s^lat_I$ with their properties.

## Key findings

- $s^lat$ does not always agree with $s$
- $s^lat$ is not additive under connected sums
- New invariants $s^lat_\pm$ and $s^lat_I$ are constructed and analyzed.

## Abstract

Kronheimer and Mrowka introduced a new knot invariant, called $s^\sharp$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi-positive knots with non-trivial right handed torus knots. These computations reveal some unexpected phenomena: we show that $s^\sharp$ does not have to agree with $s$, and that $s^\sharp$ is not additive under connected sums of knots. Inspired by our computations, we separate the invariant $s^\sharp$ into two new invariants for a knot $K$, $s^\sharp_+(K)$ and $s^\sharp_-(K)$, whose sum is $s^\sharp(K)$. We show that their difference satisfies $0 \leq s^\sharp_+(K) - s^\sharp_-(K) \leq 2$. This difference may be of independent interest. We also construct two link concordance invariants that generalize $s^\sharp_\pm$, one of which we continue to call $s^\sharp_\pm$, and the other of which we call $s^\sharp_I$. To construct these generalizations, we give a new characterization of $s^\sharp$ using immersed cobordisms rather than embedded cobordisms. We prove some inequalities relating the genus of a cobordism between two links and the invariant $s^\sharp$ of the links. Finally, we compute $s^\sharp_\pm$ and $s^\sharp_I$ for torus links.