Contact (+1)-surgeries along Legendrian Two-component Links

TL;DR

This paper investigates contact (+1)-surgeries on Legendrian two-component links in the standard contact 3-sphere, demonstrating vanishing invariants and overtwistedness using algebraic and contact-geometric methods.

Contribution

It introduces new algebraic and geometric techniques to analyze contact (+1)-surgeries on Legendrian links, including a link surgery formula for Heegaard Floer homology.

Findings

Vanishing of contact Ozsváth-Szabó invariant for certain surgeries

Overtwistedness of resulting contact 3-manifolds in specific configurations

Application of link surgery formula to contact topology

Abstract

In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsv\'{a}th-Szab\'{o} invariant for contact $(+1)$-surgery along certain Legendrian two-component links. The main tool is a link surgery formula for Heegaard Floer homology developed by Manolescu and Ozsv\'{a}th. On the other hand, we use contact-geometric argument to show the overtwistedness of the contact 3-manifolds obtained by contact $(+1)$-surgeries along Legendrian two-component links whose two components are linked in some special configurations.