All Questions
Tagged with triangulations co.combinatorics
23 questions
29
votes
1
answer
2k
views
High-Dimensional Analogs of Polygon Spaces
[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.]
Background: Polygon spaces
Given a ...
12
votes
4
answers
2k
views
What is the number of equitriangulations of the n-cube?
I wonder if this question has been considered before and if anything is known. My search attempts have failed so far.
Let's consider the n-dimesnional cube, [0,1]n, and let's call a simplex with ...
12
votes
0
answers
229
views
3-manifolds with stacked links
Stacked spheres
A triangulation of a 2-dimensional sphere is called a stacked sphere if it is obtained inductively from the boundary of a 3-simplex by deleting a 2-face (triangle) T adding a new ...
9
votes
1
answer
793
views
Properties a triangulation must have in order to describe a manifold
I am mainly interested in the 3-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every 3-dimensional topological manifold (with ...
9
votes
1
answer
424
views
Hamiltonian circuit
Let us consider a disk with one labelled point on the boundary and n labelled points in the interior.
Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...
9
votes
0
answers
212
views
Left adjoint functor between categories of polygons?
EDIT: Based on very helpful comments from Alec Rhea and Qiaochu Yuan I am adding some specification on objects and morphisms, hoping that this clarifies the idea behind these categories. I have also ...
8
votes
2
answers
850
views
Three-dimensional triangulations with fixed number of vertices
My question is the following:
Are there triangulations of S3 which (a) are non-degenerate, (b)
have four vertices, and (c) have no edges of degree two?
A side question:
If one represents this ...
8
votes
1
answer
618
views
When is a triangulation of sphere two-colorable?
Let T be a triangulation of sphere. We say that T is k-colorable if the triangles of T can be assigned with k colors such that any two triangles with a common edge have different colors.
I ...
8
votes
3
answers
921
views
Alternating Sum Involving Catalan Numbers
I was wondering if anyone knew how to obtain a simpler closed form of the following sum(or had any other insights regarding it):
\sum_{k=0}^n (-1)^k{n \choose k} C_{2n-2-k}
Here $C_n = \frac{1}{n+...
7
votes
1
answer
186
views
How many maximal triangulations of a rectangle?
I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow.
Let L_{m,n}\subset\mathbb R^2 be a rectangle given by [0,m]×[0,n] with $m,...
6
votes
2
answers
159
views
Eberhard-type theorems for Fisk triangulations?
A triangulation of a surface is called a Fisk triangulation if the degree of all but two vertices is even, and these two exceptional vertices of odd degree are neighbors.
I would like to know what ...
5
votes
1
answer
261
views
Do random triangulation edge-flips maintain randomness?
Let S be a fixed set of n points in the plane in general position.
Let T be a triangulation
of S, (somehow) selected
uniformly at random from all triangulations of S.
(There are an ...
4
votes
1
answer
255
views
Is every (not necessarily PL-) triangulation of a manifold pure, non-branching and strongly-connected?
A triangulation of a topological manifold \mathcal{M} possibly with boundary is an abstract simplicial complex \Delta together with a homeomorphism \varphi:\vert\Delta\vert\to\mathcal{M}, where $...
4
votes
1
answer
172
views
The number of simplicial and general d-polytopes with d+3 labelled vertices
Micha Perles used Gale diagrams to compute the number of simplicial d-polytopes with d+3 vertices and of general d-polytopes with d+3 vertices. The computation can be found in Chapter 6.3 of ...
4
votes
0
answers
286
views
Mean number of n-simplices per (n-2)-simplex in a triangulated n-manifold
Work by Tamura (extending results by Luo and Stong) shows the following.
Theorem: For any closed 3-manifold M and any rational number 4.5 < r < 6 there is a triangulation T of M for ...
3
votes
1
answer
445
views
Dehn-Sommerville relations for \Delta-complexes
Let M be a closed, triangulated manifold of dimension m and K(M) be its triangulation. Let f_i denote the number of i-simplices of K(M). As proved by Klee the face numbers satisfy the ...
3
votes
0
answers
93
views
Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
3
votes
0
answers
387
views
regular triangulations of the product of two simplices
Is description of all regular triangluations of \Delta^n\times \Delta^k known? (Regular triangulations are those which correspond to vertices of Gelfand--Kapranov--Zelevinsky secondary polytope, or, ...
2
votes
0
answers
87
views
Existence of a "generic enough" lattice point interior to a lattice triangle
Let T be a lattice triangle in \Bbb R^2 (i.e. the convex hull of three noncolinear points in \Bbb Z^2), and assume it has at least one interior lattice point. Is it always possible to find a ...
1
vote
0
answers
84
views
Number of polyhedral covers of a triangulation of S^2
For a given triangulation (combinatorial Type I. or Type II.) of a 2-sphere, what is the number of unique polygonal covers with n polygons where (n goes from 2 to N)?
Under polygonal cover, ...
1
vote
0
answers
247
views
dissections and vertices of non-convex polytopes
Let us call a finite union P of n-dimensional compact convex polytopes in \mathbb{R}^n a non-convex polytope. Recall that a dissection of P is a finite collection T of n-dimensional ...
0
votes
0
answers
98
views
Number of tetrahedra inside a sphere with boundary A
I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many ...
0
votes
0
answers
125
views
Naming convention for different type of triangulations
When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general ...