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

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.