Rectangular diagrams of surfaces: the basic moves

TL;DR

This paper introduces a Reidemeister type theorem for rectangular diagrams of surfaces, enhancing the understanding of isotopy classes of surfaces in the three-sphere and their applications to Legendrian knots.

Contribution

It establishes a set of basic moves that generate all isotopies of rectangular diagrams of surfaces, providing a foundational tool for surface and knot theory.

Findings

Proved a Reidemeister type theorem for rectangular surface diagrams

Demonstrated the usefulness of the formalism in Legendrian knot comparison

Provided a systematic method for surface isotopy classification

Abstract

In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to prove a Reidemeister type theorem for rectangular diagrams of surfaces.