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On Legendrian Thurston-Bennequin-symmetrical graphs

Trung Chau, Tanushree Shah

Abstract

This article reviews the development of Legendrian graph theory in the standard contact 3-sphere ($S^3, ξ_{std}$). 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.

On Legendrian Thurston-Bennequin-symmetrical graphs

Abstract

This article reviews the development of Legendrian graph theory in the standard contact 3-sphere (). 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.

Paper Structure

This paper contains 16 sections, 16 theorems, 23 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Generalizing Knots to Spatial Graphs
  3. Statement of Main Results
  4. Structure of the Paper
  5. Preliminaries and Background
  6. The Standard Contact Space and Legendrian Embeddings
  7. Generalized Legendrian Reidemeister Moves
  8. Thurston-Bennequin Invariant
  9. Ribbon Structures and Vertex Rigidities
  10. Minimal embeddings
  11. Legendrian Approximation and Boundedness of Invariants
  12. Minimal embeddings
  13. Combinatorial Rigidity and the $K_4$ Minor Obstruction
  14. The Thurston-Bennequin number of a Legendrian $(r,s)$-Thurston-Bennequin-symmetrical graph
  15. More (almost)-TB-symmetrical graphs, and arc transitivity
  16. ...and 1 more sections

Key Result

Theorem A

Let $\tilde{G}$ be a Legendrian embedding of a finite simple graph $G$ in $(\mathbb{R}^3,\zeta_{std})$. Let $s\geq 3$ be an integer such that $G$ has an $s$-cycle. If $G$ is TB-symmetrical, then

Figures (5)

  • Figure 1: Positive and Negative crossings
  • Figure 2: Right and Left cusp
  • Figure 3: Generalized Legendrian Reidemeister Moves
  • Figure 4: Legendrian K4 graph
  • Figure 5: A complete tripartite graph $K_{2,2,2}$, a Heawood graph, a Petersen graph, and a cube graph $Q_3$, from left to right, from top to bottom, respectively.

Theorems & Definitions (33)

  • Theorem A: Theorem \ref{['thm:TB-multiple-sum']}
  • Theorem B: Theorem \ref{['thm:2-arc']}
  • Conjecture 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 23 more