2019/08/14 by Sherry Gong, Gong, Sherry · 1 citation
Mathematics · Medicine · #Bone health and treatments #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1908.05018
openalex publication_date 2019/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≤ s^\sharp+(K) - s^\sharp-(K) ≤ 2. This difference may be of independent interest. We also construct two link concordance invariants that generalize s^\sharp_±, one of which we continue to call s^\sharp_±, and the other of which we call s^\sharpI. 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_± and s^\sharpI for torus links.