The maximal degree of the Khovanov homology of a cable link

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 ($2k+1$, $(2k+1)n$)-torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the ($2k+1$, $(2k+1)n$)-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 ($p$, $pn$)-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.Comment: 40 pages, 26 figures, I shorten some proof