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 $tb(L)\le -2$, and $|rot(L)|\ge 2g(L)+tb(L)$, then contact $(+1)$-surgery on $L$ is overtwisted.

arXiv comment41 pages, 11 figures; better explanations, mistakes corrected

References in corpus (8)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• On the classification of tight contact structures I
• Lectures on open book decompositions and contact structures
• Contact homology and one parameter families of Legendrian knots
• Legendrian and transverse cables of positive torus knots
• Non-looseness of non-loose knots
• Lectures on Contact Geometry in Low-Dimensional Topology
• Open book decompositions of fibre sums in contact topology

Cited by in corpus (5)

• The Legendrian knot complement problem
• Computing the Thurston-Bennequin invariant in open books
• Contact Surgeries on the Legendrian Figure-Eight Knot
• Cosmetic contact surgeries along transverse knots and the knot complement problem
• Computing rotation numbers in open books