Rectangular diagrams of surfaces: representability

TL;DR

This paper introduces rectangular diagrams of surfaces in the three-sphere, linking them to contact structures and Legendrian knots, and investigates which surface isotopy classes can be represented by such diagrams.

Contribution

It develops a combinatorial method to represent surfaces in S^3 via rectangular diagrams and explores their relation to contact structures and Legendrian knots.

Findings

No restriction on the isotopy class of the surface

Restriction exists on the boundary link's rectangular diagram

Constructs an annulus as a potential counterexample to a Legendrian knot conjecture

Abstract

We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on $\mathbb S^3$ and to rectangular diagrams of links. By using rectangular diagrams of surfaces we are going, in particular, to develop a method to distinguish Legendrian knots. This requires a lot of technical work of which the present paper addresses only the first basic question: which isotopy classes of surfaces can be represented by a rectangular diagram. Vaguely speaking the answer is this: there is no restriction on the isotopy class of the surface, but there is a restriction on the rectangular diagram of the boundary link that can arise from the presentation of the surface. The result extends to Giroux's convex surfaces for which this restriction on the boundary has a natural meaning. In a subsequent paper we are going to consider transformations of rectangular diagrams of surfaces and to study their properties. By using the formalism of rectangular diagrams of surfaces we also produce here an annulus in $\mathbb S^3$ that we expect to be a counterexample to the following conjecture: if two Legendrian knots cobound an annulus and have zero Thurston--Bennequin numbers relative to this annulus, then they are Legendrian isotopic.