Legendrian lens space surgeries

TL;DR

This paper demonstrates that all tight contact structures on certain lens spaces can be constructed via a single Legendrian surgery on a specific negative torus knot in the 3-sphere, unifying their construction.

Contribution

It provides a unified method to realize all tight contact structures on specified lens spaces through Legendrian surgeries on a particular class of knots.

Findings

All tight contact structures on lens spaces $L(ns^2-s+1,s^2)$ are obtainable via Legendrian surgery.

A single Legendrian surgery suffices for these constructions.

The method applies to both tight and overtwisted contact structures on the 3-sphere.

Abstract

We show that every tight contact structure on any of the lens spaces $L(ns^2-s+1,s^2)$ with $n\geq 2$, $s\geq 1$, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1))$ in the tight or an overtwisted contact structure on the 3-sphere.