Transverse invariants from Khovanov-type homologies

Abstract

In this paper, we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan’s deformation. These invariants, called [Formula: see text]-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar’s invariants [Formula: see text]. From the [Formula: see text]-invariants, we extract two non-negative integers which are transverse invariants (the [Formula: see text]-invariants). Finally, we give several conditions which imply the non-effectiveness of the [Formula: see text]-invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant [Formula: see text], and the non-effectiveness of the vanishing of [Formula: see text], for all prime knots with less than 12 crossings.