A recursive formula for the Khovanov cohomology of Kanenobu knots
Bull. Korean Math. Soc. 2017 Vol. 54, No. 1, 1-15
Published online December 21, 2016
Printed January 31, 2017
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
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