On Legendrian Thurston-Bennequin-symmetrical graphs

TL;DR

This paper develops a criterion for computing the Thurston-Bennequin invariant of Legendrian graphs in the standard contact 3-sphere, verified for small graphs and extended to infinite families, highlighting symmetry properties.

Contribution

It introduces a generalized criterion for calculating the total Thurston-Bennequin invariant of Legendrian graphs, applicable to symmetric graphs with up to 9 vertices and beyond.

Findings

The criterion allows computation of the total Thurston-Bennequin invariant from smaller cycles.

Verified the criterion for graphs with up to 9 vertices.

Constructed infinite families of graphs where the criterion holds.

Abstract

This article reviews the development of Legendrian graph theory in the standard contact 3-sphere ($S^3, \xi_{std}$). We provide a generalized criterion under which the total Thurston-Bennequin invariant of a Legendrian graph (sum of tb of all cycles of the Legendrian graph) can be computed from the tb of its smaller cycles. We verify this criterion for graphs with up to 9 vertices and construct infinite families of examples where it holds. We also present examples demonstrating that each condition in the criterion is necessary. Notably, the graphs satisfying this criterion exhibit a high degree of symmetry.