Ozsv\'ath-Szab\'o invariants of contact surgeries

TL;DR

This paper develops new criteria for tightness in positive contact surgeries on knots in the 3-sphere, using Floer-theoretic invariants, and computes Ozsváth-Szabó invariants for these surgeries, extending previous results.

Contribution

It introduces new tightness criteria for positive surgeries and provides explicit computations of Ozsváth-Szabó invariants for contact surgeries along Legendrian knots.

Findings

New tightness criteria for positive surgeries

Explicit Ozsváth-Szabó invariant calculations

Extension of contact surgery results to rational surgeries

Abstract

We give new tightness criteria for positive surgeries along knots in the 3-sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matic's, Ozsvath and Szabo's Floer-theoretic contact invariants. We compute the Ozsvath and Szabo's invariant of positive contact surgeries along Legendrian knots in the 3-sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries.