Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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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.

A recursive formula for the Khovanov cohomology of Kanenobu knots

Fengchun Lei and Meili Zhang

Dalian University of Technology, Dalian Naval Academy

Abstract

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