Contact surgery and transverse invariants

TL;DR

This paper develops new results on tight contact structures on 3-manifolds obtained via surgery on knots in S^3, introducing a transverse knot invariant and extending previous bounds on Thurston-Bennequin numbers.

Contribution

It introduces a new transverse knot invariant applicable under specific conditions and extends existing results relating to maximal self-linking numbers and contact structures.

Findings

Derived new existence results for tight contact structures via surgery.

Defined a new invariant for transverse knots in contact 3-manifolds.

Extended previous bounds on Thurston-Bennequin numbers.

Abstract

We derive new existence results for tight contact structures on certain 3-manifolds which can be presented as surgery along specific knots in S^3. Indeed, we extend our earlier results on knots with maximal Thurston-Bennequin number being equal to 2g-1 to knots for which the maximal self-linking number satisfies the same equality. In the argument (using contact surgery) we define an invariant for transverse knots in contact 3-manifolds under the assumption that either the knot is null-homologous or the 3-manifold has no S^1xS^2-factor in its prime decomposition, and we study its properties using the Ozsvath-Szabo contact invariant.