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 $\left(2k+1,\left(2k+1\right)n\right)$–torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the $\left(2k+1,\left(2k+1\right)n\right)$–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 $\left(p,pn\right)$–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.