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

The Rasmussen Invariant, Four-genus, and Three-genus of an Almost Positive Knot Are Equal

“…As an application of Proposition 2.2, the author [19,Corollary 5.3] computed the Rasmussen invariant of twisted Whitehead doubles (with sufficiently many twists) of any knot.…”

A Khovanov Type Invariant Derived From an Unoriented HQFT for Links in Thickened Surfaces

“…We prove Theorem 1.5 in Section 4.4. For more on the minimal homological degree of Khovanov homology, see [8], [9] and [11]. This paper is organized as follows: In Section 2, we recall the definitions of HQFTs and some algebras introduced in [13], as well as those of unoriented HQFTs and extended crossed group algebras defined in [10].…”

“…We prove Theorem 1.5 in Section 4.4. For more on the minimal homological degree of Khovanov homology, see [8], [9] and [11].…”