Bull. Korean Math. Soc. 2017; 54(1): 1-15
Online first article December 21, 2016 Printed January 31, 2017
https://doi.org/10.4134/BKMS.b141003
Copyright © The Korean Mathematical Society.
Fengchun Lei and Meili Zhang
Dalian University of Technology, Dalian Naval Academy
Kanenobu has given infinite families of knots with the same HOMFLY polynomial invariant but distinct Alexander module structure. In this paper, we give a recursive formula for the Khovanov cohomology of all Kanenobu knots $K(p,q)$, where $p$ and $q$ are integers. The result implies that the rank of the Khovanov cohomology of $K(p,q)$ is an invariant of $p+q$. Our computation uses only the basic long exact sequence in knot homology and some results on homologically thin knots.
Keywords: homologically thin knot, Jones polynomial, signature, Kanenobu knots, Khovanov cohomology, odd Khovanov homology
MSC numbers: 57M25, 57M27
2017; 54(6): 2149-2154
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