Classification of invariant tight contact structures on the 3-space, -ball and -sphere

TL;DR

This paper classifies invariant tight contact structures on 3-space, ball, and sphere under involutions, revealing new torsion phenomena that influence their classification and potentially aiding in the study of invertible Legendrian knots.

Contribution

It introduces a classification framework for invariant tight contact structures considering involutions, highlighting the role of new integral torsion in their distinctions.

Findings

Identification of new integral torsion affecting classification

Splitting of equivalence classes due to torsion

Potential applications to Legendrian knot classification

Abstract

We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the classical scenario, a new integral torsion appears, dictating a splitting between equivalence classes. These tools could be useful fur future classification results regarding strongly invertible Legendrian knots.