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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.
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참고문헌 (26)
참고문헌 신청
A. Lobb / 2014 / The Kanenobu knots and Khovanov-Rozansky homology / Proc. Amer. Math. Soc 142 (4) : 1447 ~ 1455
A. Shmakovitch / / A program for computing khovanov homology
C. Manolescu / 2007 / On the Khovanov and knot Floer homologies of quasi-alternating links / proceedings of the 14th Gokova Geometry-Topology Conference : 60 ~ 81
D. Bar-Natan / / The knot atlas
D. Bar-Natan / 2002 / On Khovanov’s categorification of the Jones polynomial / Algebr. Geom. Topol 2 : 337 ~ 370
A. Shmakovitch / / A program for computing khovanov homology
C. Manolescu / 2007 / On the Khovanov and knot Floer homologies of quasi-alternating links / proceedings of the 14th Gokova Geometry-Topology Conference : 60 ~ 81
D. Bar-Natan / / The knot atlas
D. Bar-Natan / 2002 / On Khovanov’s categorification of the Jones polynomial / Algebr. Geom. Topol 2 : 337 ~ 370
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