On the Thurston–Bennequin invariant of graph divide links

Abstract

We determine the Thurston-Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A. I. Stipsicz, we prove that the 3-manifold obtained from S 3 by Dehn surgery along a non-trivial graph divide knot K with coefficient r carries positive, tight contact structures for every r except the Thurston-Bennequin invariant of K.

On the Lagrangian fillability of almost positive links

“…Tomomi Kawamura taught the author that this fact had been proved by Mikami Hirasawa (see also [25, Remark 6.9]). Ishikawa [23] proved that the maximal Thurston-Bennequin number of any graph divide link is equal to its slice Euler characteristic. This means graph divide links satisfy a necessary condition to be Lagrangian fillable.…”

Contact structures and reducible surgeries

“…Thus in each of these cases the result follows from Proposition 3.2. The remaining "sporadic" knots, families IX-XII, have tb = 2g − 1 because they are all divide knots [79], hence they satisfy tb(K) = 2g s (K) − 1 [42], and as L-space knots they have g s (K) = g(K) [64], and the result follows.…”