The maximal degree of the Khovanov homology of a cable link

Tagami, Keiji

doi:10.2140/agt.2013.13.2845

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The maximal degree of the Khovanov homology of a cable link

Keiji Tagami

Algebraic & Geometric Topology 13 (2013) 2845–2896

DOI: 10.2140/agt.2013.13.2845

Abstract

In this paper, we study the Khovanov homology of cable links. We first estimate the maximal homological degree term of the Khovanov homology of the $(2 k + 1, (2 k + 1) n)$ –torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the $(2 k + 1, (2 k + 1) n)$ –torus link is greater than or equal to $k^{2} n + 2$ . Next, we study the maximal homological degree of the Khovanov homology of the $(p, p n)$ –cabling of any knot with sufficiently large $n$ . Furthermore, we compute the maximal homological degree term of the Khovanov homology of such a link with even $p$ . As an application we compute the Khovanov homology and the Rasmussen invariant of a twisted Whitehead double of any knot with sufficiently many twists.

Keywords

knot, Khovanov homology, cable link, Rasmussen invariant

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57M25

References

Publication

Received: 23 October 2012

Revised: 16 March 2013

Accepted: 17 April 2013

Published: 20 July 2013

Authors

Keiji Tagami
Department of Mathematics Tokyo Institute of Technology Ookayama, Meguro Tokyo 152-8551 Japan

Volume 13, issue 5 (2013)