#### Volume 17, issue 3 (2017)

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The intersection graph of an orientable generic surface

### Doron Ben Hadar

Algebraic & Geometric Topology 17 (2017) 1675–1700
##### Abstract

The intersection graph $M\left(i\right)$ 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 branch values. $M\left(i\right)$ has an enhanced graph structure which Gui-Song Li referred to as a “daisy graph”. If $F$ is oriented, then the orientation further refines the structure of $M\left(i\right)$ into what Li called an “arrowed daisy graph”.

Li left open the 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.

##### Keywords
generic surfaces, immersed surfaces in 3-manifolds, intersection graph
##### Mathematical Subject Classification 2010
Primary: 57N10, 57N12
Secondary: 57N35, 57N40, 57N75
##### Publication
Received: 20 January 2016
Accepted: 2 July 2016
Published: 17 July 2017
##### Authors
 Doron Ben Hadar Department of Mathematics Bar-Ilan University 5290002 Ramat Gan Israel http://math.biu.ac.il/node/641