Neighborly 2-Manifolds with 12 Vertices

Abstract

We provide a complete list of 59 orientable neighborly 2-manifolds with 12 vertices of genus 6, and we study their possible flat embeddings in Euclidean 3-space. Whereas the question of embeddability remains open in its general form, we obtain several properties of the embedding (polyhedral realization) under the assumption that it does exist:1. The order of the geometrical automorphism group of any polyhedral realization would not exceed 2.2. The polyhedral realization would not be obtainable via the Schlegel diagram of any 4-polytope; moreover, none of our orientable neighborly 2-manifolds with 12 vertices can be found within of the 2-skeleton of any 4-polytope.3. The polyhedral realization would not allow a tetrahedral subdivision without inserting new vertices.By using a weaker version of the manifold property, we obtain neighborly polyhedra with 2n vertices for every n 3.

Degree-regular triangulations of the double-torus

“…Observe that G 5 (EG(N 5 )) = C 4 (0, 3, 10, 5) ∪ C 4 (4, 7, 2, 9) ∪ C 4 (8,11,6,1), E 2,5 = {{0, 2}, {4, 6}, {8, 10}} and E 6,5 = {{0, 10}, {2, 4}, {6, 8}}.…”

“…Now, completing successively, we get lk(8) = C 7 (2, 1, 9, 4, u, 5, v) and lk(v) = C 7 (2, 7, u, 9, 4, 5, 8). Then M ∼ = N 1 by the map ψ • (1,8,7,4) (2, 6, u, 3)(5, v). Subcase 1.2.…”

“…It is easy to see that x = 8 or u. If x = 8 then, lk(3) = C 7 (4,0,2,9,1,7,8). Now, lk(2) = C 7 (9, 3, 0, 1, 8, y, z), for some y, z ∈ V .…”

“…Formally, |X| is the subspace of [0, 1] V (X) consisting of the functions f : V (X) → [0,1] such that the support {v ∈ V (X) : f (v) = 0} is a simplex of X and v∈V (X) f (v) = 1. If σ is a simplex then |σ| := {f ∈ |X| : v∈σ f (v) = 1} is called the geometric simplex corresponding to σ.…”

Torus Actions and Their Applications in Topology and Combinatorics

“…Triangulated (or simplicial) manifolds are the subject of section 2. 4; some related open problems from low-dimensional and PL topology are also included there. The notion of bistellar moves, as a particularly interesting and useful class of operations on simplicial complexes, is discussed in section 2.5.…”