Rectangular Diagrams of Legendrian Graphs

TL;DR

This paper introduces a combinatorial approach to classifying Legendrian graphs using rectangular diagrams and establishes a one-to-one correspondence with fence diagrams, simplifying their study.

Contribution

It develops a new combinatorial model for Legendrian graphs and proves their classification aligns with fence diagrams, advancing understanding in contact topology.

Findings

Rectangular diagrams represent Legendrian graphs combinatorially.

Equivalence of Legendrian graphs corresponds to rectangular diagram moves.

Legendrian graph classes are in one-to-one correspondence with fence diagrams.

Abstract

In this paper Legendrian graphs in $(\mathbb{R}^3,\xi_{\mathrm{st}})$ are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram --- a purely combinatorial object. Moves of rectangular diagrams are introduced so that equivalence classes of Legendrian graphs and rectangular diagrams coincide. Using this result we prove that the classes of Legendrian graphs are in one-to-one correspondence with fence diagrams modulo fence moves introduced by Rudolph.