Khovanov homology and the symmetry group of a knot

Watson, L. (2017) Khovanov homology and the symmetry group of a knot. Advances in Mathematics, 313, pp. 915-946. (doi: 10.1016/j.aim.2017.04.003)

139229.pdf - Accepted Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.



We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial. While closely tied to Khovanov homology — and hence the Jones polynomial — we observe that this new invariant detects non-amphicheirality in subtle cases where Khovanov homology fails to do so. In fact, we exhibit examples of knots that are not distinguished by Khovanov homology but, owing to the presence of a strong inversion, may be distinguished using our invariant. This work suggests a strengthened relationship between Khovanov homology and Heegaard Floer homology by way of two-fold branched covers that we formulate in a series of conjectures.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Watson, Professor Liam
Authors: Watson, L.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Mathematics
ISSN (Online):1090-2082
Published Online:16 May 2017
Copyright Holders:Copyright © 2017 Elsevier Inc.
First Published:First published in Advances in Mathematics 313: 915-946
Publisher Policy:Reproduced in accordance with the publisher copyright policy
Related URLs:

University Staff: Request a correction | Enlighten Editors: Update this record

Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
651711HFFUNDGRP: New connections in low-dimensional topology: Relating Heegaard Floer homology and the fundamental groupLiam WatsonEuropean Commission (EC)631364M&S - MATHEMATICS