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Stability and triviality of the transverse invariant from Khovanov homology. (English) Zbl 1440.57004
Summary: We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by O. Plamenevskaya [Math. Res. Lett. 13, No. 4, 571–586 (2006; Zbl 1143.57006)] for their closures, which are naturally transverse links in the standard contact 3-sphere. For any 3-braid \(\beta \), we show that the transverse invariant of its closure does not vanish whenever the fractional Dehn twist coefficient of \(\beta\) is strictly greater than one. We show that Plamenevskaya’s transverse invariant is stable under adding full twists on \(n\) or fewer strands to any \(n\)-braid, and use this to detect families of braids that are not quasipositive. Motivated by the question of understanding the relationship between the smooth isotopy class of a knot and its transverse isotopy class, we also exhibit an infinite family of pretzel knots for which the transverse invariant vanishes for every transverse representative, and conclude that these knots are not quasipositive.
MSC:
57K10 Knot theory
57R17 Symplectic and contact topology in high or arbitrary dimension
57K18 Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.)
Software:
KnotInfo
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