On the Kronheimer–Mrowka concordance invariant

DOI10.1112/TOPO.12175zbMATH Open1477.57014arXiv1908.05018OpenAlexW3106661238MaRDI QIDQ4999726

Sherry Gong

Publication date: 2 July 2021

Published in: Journal of Topology (Search for Journal in Brave)

Abstract: Kronheimer and Mrowka introduced a new knot invariant, called $s^{s}harp$, 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 $s^{s}harp$ does not have to agree with $s$, and that $s^{s}harp$ is not additive under connected sums of knots. Inspired by our computations, we separate the invariant $s^{s}harp$ into two new invariants for a knot $K$, $s^{s}har{p}_{+}(K)$ and $s^{s}har{p}_{-}(K)$, whose sum is $s^{s}harp(K)$. We show that their difference satisfies $0leq{s}^{s}har{p}_{+}(K)-s^{s}har{p}_{-}(K)leq2$. This difference may be of independent interest. We also construct two link concordance invariants that generalize $s^{s}har{p}_{p}m$, one of which we continue to call $s^{s}har{p}_{p}m$, and the other of which we call $s^{s}har{p}_I$. To construct these generalizations, we give a new characterization of $s^{s}harp$ using immersed cobordisms rather than embedded cobordisms. We prove some inequalities relating the genus of a cobordism between two links and the invariant $s^{s}harp$ of the links. Finally, we compute $s^{s}har{p}_{p}m$ and $s^{s}har{p}_I$ for torus links.

Full work available at URL: https://arxiv.org/abs/1908.05018

zbMATH Keywords

Rasmussen invariantKronheimer-Mrowka concordance invariant

Mathematics Subject Classification ID

General geometric structures on low-dimensional manifolds (57M50) Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) (57K18) Invariants of 4-manifolds (including Donaldson and Seiberg-Witten invariants) (57K41)

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