Transverse Surgery on Knots in Contact 3-Manifolds

Abstract

We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. The overarching theme of this paper is to show that in many contexts, surgery on transverse knots is more natural than surgery on Legendrian knots. Besides reinterpreting surgery on Legendrian knots in terms of transverse knots, our main results on are in two complementary directions: conditions under which inadmissible transverse surgery (\textit{cf.\@} positive contact surgery on Legendrian knots) preserves tightness, and conditions under which it creates overtwistedness. In the first direction, we give the first result on the tightness of inadmissible transverse surgery for contact manifolds with vanishing Heegaard Floer contact invariant. In particular, inadmissible transverse surgery on the connected binding of a genus $g$ open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than $2g-1$. In the second direction, along with more general statements, we deduce a partial generalisation to a result of Lisca and Stipsicz: when $L$ is a Legendrian knot with $tb(L) \leq -2$, and $|rot(L)| \geq 2g(L)+tb(L)$, then contact $(+1)$-surgery on $L$ is overtwisted.