Rectangular diagrams of surfaces: distinguishing Legendrian knots

TL;DR

This paper develops a combinatorial approach using rectangular and mirror diagrams to distinguish Legendrian knots, introducing moves and their commutation properties, and applying these to identify non-equivalent Legendrian knots with the same topological type.

Contribution

It introduces new diagrammatic moves and their commutation relations, providing a novel method for distinguishing Legendrian knots based on combinatorial surface representations.

Findings

Established moves that relate isotopic surface presentations

Proved commutation relations between move types

Applied the method to distinguish specific Legendrian knots

Abstract

In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce moves that allow transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams and various moves for them that can be used to transform presentations of isotopic surfaces to each other. The moves can be divided into two types so that, vaguely speaking, type~I moves commute with type~II ones. This commutation is the matter of the main technical result of the paper. We use it as well as a relation of the moves to Giroux's convex surfaces to propose a new method for distinguishing Legendrian knots. We apply this method to show that two Legendrian knots having topological type $6_2$ are not equivalent. More applications of the method will be a subject of subsequent papers.