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

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.