Normal surfaces as combinatorial slicings

Abstract

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a lower bound on the number of quadrilaterals of normal surfaces depending on the genus g is presented. It is shown to be sharp for infinitely many values of g. Furthermore we classify slicings of combinatorial 3-manifolds with a maximum number of edges in the slicing.Comment: 18 pages, 9 figure

Bounds for the genus of a normal surface

“…Examples of normal surfaces S can be constructed with q(S) = 3g(S) + O( g(S)) quadrilaterals (see Example 4 for a discussion of these examples). All these examples are well within the conjectured bound of q(S) ≥ 3g(S) from[42].Corollary 15 (Bound for normal surface in minimal, prime manifold) Let M be a triangulated, compact, orientable, prime 3-manifold, and S be a closed, connected, orientable normal surface in M. If the triangulation of M is minimal, then…”

Combinatorial 3-Manifolds with Transitive Cyclic Symmetry

“…The pre-image of a point under an rsl-function which does not meet any vertex of the surrounding combinatorial manifold is called a slicing. By construction, a slicing is an embedded co-dimension 1 submanifold which contains information about the topology of the surrounding manifold (see Figures 4.1, 4.2 and 5.2 for slicings in the case d = 3, and [42] for further details about slicings).…”

Simplicial blowups and discrete normal surfaces in simpcomp

“…• Support for discrete normal surfaces [17,16,28] and slicings: slicings of combinatorial d-manifolds are (non-singular) (d − 1)-dimensional level sets of polyhedral Morse functions. In dimension 3, slicings are discrete normal surfaces.…”

“…For n ≥ 3, consider the cyclic 4-polytope C 4 (2n) on 2n vertices with vertex labels 1 to n. By Gale's evenness condition, neither the span of all odd nor the span of all even vertices in C 4 (2n) contains a triangle of C 4 (2n). Thus, given the combinatorial 3-sphere S = ∂C 4 (2n), a level set of a Morse function on S separating the even from the odd vertices gives rise to a handle body decomposition of S -this is a discrete normal surface in the sense of [28]. This construction can be done in simpcomp as follows.…”