The intersection graph of an orientable generic surface

TL;DR

This paper characterizes which arrowed daisy graphs can be realized as intersection graphs of oriented generic surfaces in 3D space, answering an open question from prior research.

Contribution

It provides a complete characterization of realizable arrowed daisy graphs for oriented generic surfaces, extending previous work on intersection graphs.

Findings

Provides necessary and sufficient conditions for realization

Answers the open question posed by Li in 1998

Includes generalizations and extensions to the main theorem

Abstract

I answer an open question left by Gui-Song Li in "On self-intersections of immersed surfaces" (AMS Proceedings, Volume 126, 1998, pp.3721-3726.) The intersection graph $M(i)$ of a generic surface $i:F \to S^3$ is the set of values which are either singularities or intersections. It is a multigraph whose edges are transverse intersections of two surfaces and whose vertices are triple intersections and cross-caps. $M(i)$ has an additional structure which Li called "a daisy graph." If F is oriented then the orientation further refines $M(i)$'s structure into what Li called an "arrowed daisy graph." Li left the open question "which arrowed daisy graphs can be realized as the intersection graph of an oriented generic surface?" The main theorem of this article will answer this. I will also provide some generalizations and extensions to this theorem in sections 4 and 5.