A diagrammatic presentation and its characterization of non-split compact surfaces in the 3-sphere

TL;DR

This paper introduces a diagrammatic method for representing non-split compact surfaces in the 3-sphere using signed spatial graphs, and characterizes their isotopy classes via Reidemeister moves.

Contribution

It provides a new diagrammatic presentation and a complete characterization of ambient isotopy for such surfaces in the 3-sphere.

Findings

Reidemeister moves correspond to isotopies of surfaces

Diagrammatic presentation captures surface embeddings accurately

Characterization of surface isotopy via diagram transformations

Abstract

We give a presentation for a non-split compact surface embedded in the 3-sphere by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces represented by the diagrams are ambient isotopic in the 3-sphere.