Divisibility of Lee's class and its relation with Rasmussen's invariant

Abstract

Lee homology (a variant of Khovanov homology) over $\mathbb{Q}$ possesses the "canonical generators" as its basis. The generators (Lee's classes) $[\alpha (D,o)]$ are constructed combinatorially from an oriented link diagram $D$, one for each alternative orientation $o$ on $D$. Let $R$ be an integral domain. There exists a family of link homology theory $\{H_c(-;R){\}}_{c\in R}$, where Khovanov's theory corresponds to $c=0$ and Lee's theory corresponds to $c=2$. For each $c\in R\setminus 0$, Lee's classes $[\alpha (D,o)]$ can be defined as elements in $H_c(D;R)$, but when $c$ is not invertible then they do not form a basis; in fact they are divisible by $c$-powers. We define the $c$-divisibility $k_c(D)$ of $[\alpha (D,o)]$ with $o$ the given orientation of $D$. For any link $L$ and its diagram $D$, we prove that ${\bar{s}}_c(L):=2k_c(D)+w(D)-r(D)+1$ is a link invariant, where $w$ is the writhe, and $r$ is the number of Seifert circles. We pose the question whether ${\bar{s}}_c$ coincides with Rasmussen's $s$-invariant. There are several evidences that support the affirmative answer. For instance, ${\bar{s}}_c$ is a link concordance invariant, and the Milnor conjecture can be reproved using ${\bar{s}}_c$. Also for the special case $(R,c)=(\mathbb{Q}[h],h)$, our ${\bar{s}}_c$ actually coincides with $s$ as knot invariants.

This paper is based on the master's thesis submitted to the Graduate School of Mathematical Sciences, the University of Tokyo

References in corpus (3)

• The algebraic crossing number and the braid index of knots and links
• The Bar-Natan homology and unknotting number
• Khovanov's invariant for closed surfaces

Cited by in corpus (5)

• Fixing the functoriality of Khovanov homology: a simple approach
• A family of slice-torus invariants from the divisibility of Lee classes
• Instantons, special cycles, and knot concordance
• Involutive Khovanov homology and equivariant knots
• A Bar-Natan homotopy type