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Mathematics, Geometric Topology
Abstract:
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 Plamenevskaya 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.
