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 (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 tbpLq ď´2, and |rotpLq| ě 2gpLq`tbpLq, then contact p`1q-surgery on L is overtwisted. arXiv:1409.7077v3 [math.GT]

Cosmetic contact surgeries along transverse knots and the knot complement problem

“…There exist also the notions of admissible and inadmissible transverse surgery which give a natural choice of contact structure on the surgered manifold (depending only on the tubular neighborhood in the case of admissible transverse surgery), see [Ga99,Ga02,BaEt13,Co15] for the precise definitions. From the work of Conway [Co15, Theorem 1.10] it follows that all inadmissible transverse surgeries along transverse unknots in (S 3 , ξ st ) with self-linking number sl ≤ −2 yield overtwisted contact manifolds.…”

Symplectic Fillings, Contact Surgeries, and Lagrangian Disks

“…In these cases, we will always consider the tight contact structure on the torus with maximally negative relative Euler class. This choice agrees with those studied in the Heegaard Floer literature [28,34,36] and corresponds to inadmissible transverse surgery [6]. In the language of [11] we choose all stabilizations of the Legendrian knots to be negative.…”

“…Moreover, if one performs contact (r)-surgery on L in N , then there is a transverse knot T in the new solid torus N , and the slopes realized by characteristic foliations on boundary-parallel tori in N have slopes clockwise of r and counterclockwise of 0. If we then do admissible transverse surgery on T with slope s ∈ (r, ∞), then the resulting contact manifold is the same as the result of contact s-surgery on L, see [6].…”

On Overtwisted Contact Surgeries

“…Conjecture 6.13 from [2] states that if L is a nullhomologous Legendrian knot with tb(L) ≤ −2, then contact (+n)-surgery on L is overtwisted, for any positive integer n < |tb(L)|. We give a negative answer to this conjecture by constructing a knot with tb = −3 where (+n)-surgery along the knot is always tight for any integer n ≥ 2.…”

“…Following Lemma 6.4 of [2] which extends Lemma 2 of [10], Lemma 6.6 of [12], cf. [5], to more general contact 3-manifolds, the linking matrix M is the (n × n) matrix…”