The total Thurston-Bennequin number of complete and complete bipartite Legendrian graphs

TL;DR

This paper introduces the total Thurston-Bennequin number as a new invariant for Legendrian graphs, relating it to cycle invariants and exploring implications for specific complete and bipartite graphs.

Contribution

It defines the total Thurston-Bennequin number for Legendrian graphs and establishes its determination by cycle invariants for complete and bipartite graphs.

Findings

The total Thurston-Bennequin number is determined by 3-cycle invariants in complete graphs.

The total Thurston-Bennequin number is determined by 4-cycle invariants in complete bipartite graphs.

Implications are discussed for K_4, K_5, and K_{3,3}.

Abstract

We study the Thurston-Bennequin number of complete and complete bipartite Legendrian graphs. We define a new invariant called the total Thurston-Bennequin number of the graph. We show that this invariant is determined by the Thurston-Bennequin numbers of 3-cycles for complete graphs and by the Thurston-Bennequin number of 4-cycles for complete bipartite graphs. We discuss the consequences of these results for K_4, K_5 and K_{3,3}.