Tight contact structures on toroidal plumbed 3-manifolds

TL;DR

This paper studies tight contact structures on certain plumbed 3-manifolds, providing methods to count, construct, and modify these structures, especially focusing on Giroux torsion and Stein diagram representations.

Contribution

It introduces an explicit algorithm for constructing Stein diagrams of tight structures without Giroux torsion on plumbed 3-manifolds.

Findings

Count of tight contact structures with zero Giroux torsion provided.

Conditions for adding Giroux torsion without overtwisting analyzed.

Algorithm for constructing Stein diagrams for these structures developed.

Abstract

We consider tight contact structures on plumbed 3-manifolds with no bad vertices. We discuss how one can count the number of tight contact structures with zero Giroux torsion on such 3-manifolds and explore conditions under which Giroux torsion can be added to these tight contact structures without making them overtwisted. We give an explicit algorithm to construct stein diagrams corresponding to tight structures without Giroux torsion. We focus mainly on plumbed 3-manifolds whose vertices have valence at most 3 and then briefly consider the situation for plumbed 3-manifolds with vertices of higher valence.