A family of slice-torus invariants from the divisibility of Lee classes

TL;DR

This paper introduces a new family of slice-torus invariants derived from the divisibility of Lee classes in Khovanov homology, generalizing Rasmussen's invariant and exploring their properties and distinctions.

Contribution

It defines a new family of invariants from Lee class divisibility, proves their equivalence to Rasmussen's invariant in certain cases, and demonstrates their independence in others.

Findings

$ ilde{ss}_c$ coincides with Rasmussen's $s^F$ over fields.

$ss_c$ equals $ ilde{ss}_c$ for specific rings and elements.

$ss_3$ is not slice-torus, showing independence from Rasmussen invariants.

Abstract

We give a family of slice-torus invariants $\tilde{ss}_c$, each defined from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements $c$ in any principal ideal domain $R$. For the special case $(R, c) = (F[H], H)$ where $F$ is any field, we prove that $\tilde{ss}_c$ coincides with the Rasmussen invariant $s^F$ over $F$. Compared with the unreduced invariants $ss_c$ defined by the first author in a previous paper, we prove that $ss_c = \tilde{ss}_c$ for $(R, c) = (F[H], H)$ and $(\mathbb{Z}, 2)$. However for $(R, c) = (\mathbb{Z}, 3)$, computational results show that $ss_3$ is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.