On transverse invariants from Khovanov homology
Summary: In [Math. Res. Lett. 13, No. 4, 571--586 (2006; Zbl 1143.57006)], \textit{O. Plamenevskaya} associated to each transverse knot \(K\) an element of the Khovanov homology of \(K\). In this paper, we give two refinements of Plamenevskaya's invariant, one valued in \textit{D. Bar-Natan}'s deformation (from [Geom. Topol. 9, 1443--1499 (2005; Zbl 1084.57011)]) of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum (from [\textit{R. Lipshitz} and \textit{S. Sarkar}, J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)]). We show that the first of these refinements is invariant under negative flypes 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 refinements are determined by the classical invariants of \(K\).
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