The maximal degree of the Khovanov homology of a cable link

TL;DR

This paper investigates the maximal homological degree in Khovanov homology for cable links, providing bounds and explicit calculations for certain torus links and cablings, with applications to Whitehead doubles.

Contribution

It offers new bounds on homological thickness for specific torus links and computes maximal degrees for cable links, advancing understanding of Khovanov homology in these cases.

Findings

Homological thickness of ($2k+1$, $(2k+1)n$)-torus links is at least $k^{2}n+2$.

Maximal homological degree for ($p$, $pn$)-cabling of knots is estimated and computed for even $p$.

Khovanov homology and Rasmussen invariant are computed for twisted Whitehead doubles with many twists.

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.