Twisting, mutation and knot Floer homology
type: Item
instance of: scholarly article
title: Twisting, mutation and knot Floer homology
zbMATH Open document ID: 1431.57007
DOI: 10.4171/QT/119
author: Peter Lambert-Cole
published in: Quantum Topology
publication date: 2018-12-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1608.02011
review text: This article concerns the interesting problem of detecting knot mutations by means of knot invariants such as knot polynomials and homology theories. While many knot invariants do not detect mutations, the bigraded knot Floer homology groups can detect Conway mutation and genus $2$ mutation. The author shows that the knot Floer homology $\widehat{HFK}(\mathcal{L}_{n})$ stabilizes as $n$ goes to infinity, where $\mathcal{L}_{n}$ is a knot obtained from a given knot $\mathcal{L}$ with a fixed positive crossing by replacing this crossing with $n$ positive twists. This stabilization phenomenon categorifies a similar phenomenon of the Alexander polynomial. The author uses this property to construct an infinite family of prime, positive mutant knots with isomorphic bigraded knot Floer homology groups. This family is obtained from the Kinoshita-Terasaka knot ($11n42$) and the Conway knot ($11n34$), which are positive mutants, by adding full twists to these knots just outside the mutation sphere.
reviewed by: José Frias
Mathematics Subject Classification ID: 57K10
Mathematics Subject Classification ID: 57K18
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 6996467
zbMATH Keywords: Heegaard Floer homology
zbMATH Keywords: mutation
describes a project that uses: SnapPy
MaRDI profile type: Publication
arXiv ID: 1608.02011
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description: scientific article