Classification of tight contact structures on a solid torus

TL;DR

This paper provides a complete classification of tight contact structures on a solid torus, including a formula for counting non-isotopic structures with specified boundary dividing sets, advancing understanding in contact geometry.

Contribution

It offers a closed-form formula for the number of non-isotopic tight contact structures on a solid torus with given boundary conditions, filling a gap in prior classifications.

Findings

Derived a closed formula for classification counts

Extended classification to all boundary dividing sets

Resolved previously unknown cases in contact geometry

Abstract

It is a basic question in contact geometry to classify all non-isotopic tight contact structures on a given 3-manifold. If the manifold has a boundary, we need also specify the dividing set on the boundary. In this paper, we answer the classification question completely for the case of a solid torus by writing down a closed formula for the number of non-isotopic tight contact structures with any given dividing set on the boundary of the solid torus. Previously, only a few special cases were known due to work by Honda.