On transverse invariants from Khovanov homology

Abstract

In [31], O. Plamenevskaya associated to each transverse knot $K$ an element of the Khovanov homology of $K$. In this paper, we give two renements of Plamenevskaya’s invariant, one valued in Bar-Natan’s deformation (from [2]) of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum (from [20]). We show that the rst of these renements is invariant under negative ypes and $SZ$ moves; this implies that Plamenevskaya’s class is also invariant under these moves. We go on to show that for small-crossing transverse knots $K$, both renements are determined by the classical invariants of $K$.