On self-intersections of immersed surfaces

Abstract

Abstract. A daisy graph is a union of immersed circles in 3-space which intersect only at the triple points. It is shown that a daisy graph can always be realized as the self-intersection set of an immersed closed surface in 3-space and the surface may be chosen to be orientable if and only if the daisy graph has an even number of edges on each immersed circle.It is well known that a general position immersion of a surface in euclidean 3-space may have two types of self-intersections: double points and triple points. If the surface is closed, then its self-intersection set is a union of immersed circles, called the transverse components, which intersect only at the triple points where three arcs meet in the same way as the three axes of 3-space meet at the origin. We call such a union of immersed circles a daisy graph which is so named since it can be obtained from copies of the daisy by adding bands; see Figure 1. For our purposes, we sometimes ignore the extra configurations at the triple points and consider a daisy graph as a union of embedded circles and a standard graph in 3-space with vertices being the triple points which are of degree 6, hence the notions of vertices and edges from graph theory make sense for a daisy graph and its transverse components. But observe that embedded circles that do not pass through triple points are considered to have zero number of edges.The study of immersed surfaces in 3-space via their self-intersections goes back as far as Dehn [8] and Whitney [12]. Banchoff [1] showed that the number of triple points of an immersed closed surface in 3-space is congruent modulo two to the Euler characteristic of the surface. The proof works in any 3-manifold provided the image of the surface is null-homologous. More generally, Carter and Ko proved a congruence χ(f (M )) = χ(M ) + T (f) modulo two, where χ denotes the Euler number of a cell complex, T (f ) is the number of triple points of an immersion f of a closed surface M into an arbitrary 3-manifold N ; see [4]. Their techniques could be applied to give a proof of the result of Izumiya and Marar [10] thatwhere B(f ) is the number of branched points, and now f is a general position map. This last result is an equality rather than a congruence. Carter [2] and Csikós and Szücs [7] addressed the problem of extending an immersion of a circle in a surface to a proper immersion of a surface in a 3-manifold bounded by the surface.

The Lifting Problem is NP Complete

“…In the final chapter, 9, we will answer the question "which DADGs are realizable via an oriented generic surface in a given orientable 3-manifold M ". This is a generalization of an open question posted by Li in [13]. The answer depends solely on the first homology group H 1 (M ; Z), and whether M has a boundary.…”

“…In [13], Li left open the question "Which ADGs can be realized by a generic immersion in S 3 ". In Chapter 9, we answer a generalized version -given a 3-manifold M , which DADGs can be realized by generic surfaces in M .…”

“…In this section, we will explain this alternative invariant and how it is encoded, along with the first two invariants, in an "enriched graph structure". It will be similar to the "arrowed daisy graphs" Li defined in [13]. For algorithmic purposes, this "enriched graph structure" will be represented by a formal data type that a computer can handle.…”

“…The lifting formula of a generic surface is determined by the topology of the intersection graph of the surface and its close neighborhood. In [13] and [15], Li and Satoh (respectively) defined enriched graph structures on the intersection graph that encode the topology of its neighborhood. In chapter 7, we will use a structure very similar to Li's "arrowed daisy graphs", which we will call "digital arrowed daisy graph" (DADG), to encode this information in a more general setting.…”

Combinatorics of knotted surface diagrams

“…Banchoff's triple point formula [2] also relates singularities of surface maps to their intrinsic topology. Li shows [20] when a daisy graph is realizable as the multiple point set of a general position map. Generalizations of Banchoff's formula appear in [12] and [19].…”