# Tagged Questions

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### Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?

For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
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### Consequences of Zeeman's conjecture

Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision. Zeeman showed that this implies the Poincaré conjecture in ...
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### References to proofs of a theorem by Van Kampen-Flores

Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other. This ...
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### Existence of a sequence of (almost) Moore irregular graphs embedded on closed surfaces

Let $S_{g}$ denote the genus $g$ closed orientable surface. I'm interested in disproving the existence of a certain configuration of simple closed curves on $S_{g}$. I'd be happy to go into more ...
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### Counting edges in embeddable CW-complexes in R^3

Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...
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### How many types of jigsaw puzzle pieces in n dimensions?

I was partitioning jigsaw puzzle pieces with some friends yesterday and we noticed that there are 6 types of pieces: All 4 sides have a knobby bit sticking out 1 side has a knobby bit sticking out 2 ...
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### Right-angled polytopes

%This question is motivated by the little discussion here at the bottom. The following thing are known about hyperbolic right-angled polytopes: Compact hyperbolic right-angled polytopes do not ...
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### Reference for the image of the adjoint to the differential in graph cohomology (which yields STU & IHX)?

One can define cochain complexes of (combinatorial) graphs, where each term is a vector space of linear combinations of certain (isomorphism classes of) graphs, and where the differential $d$ is a ...
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### Classification of ribbon graphs (reference request)

Is there an explicit classification of (isomorphism classes of) ribbon graphs (fat graphs) of low genus (say 0,1,2, maybe 3) with 'few' vertices (say less than 10)? Thanks!
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### A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
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### Characterizing f-vectors of regular(Delaunay) triangulations?

The question is this; is there a complete characterization of all the f-vectors of regular triangulations? (other names for this can be: Delaunay triangulation, coherent triangulation, convex ...
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### Any map of a contractible complex to itself has a fixed point

Reading Lovasz's lecture notes on evasive graph properties, I encountered the following extension of Brouwer's fixed point theorem: Any continues map from a contractible [finite] simplicial complex ...
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### Example of collapsible complex which collapses to a non-collapsible complex

In their paper http://arxiv.org/abs/0907.2954 , Barmak and Minian claim that "there exist collapsible complexes which collapse to nontrivial subcomplexes with no free faces" but unfortunately do ...
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### Convex polyhedral decomposition of spheres

Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$? Remarks: A polyhedron is defined as an area enclosed by a ...
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### Classification of Platonic solids

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula ...
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### References for Eilenberg-Zilber shuffle product

Most of the treatments I can find in the literature for the Eilenberg-Zilber shuffle product approach it from the point of view of simplicial sets (including the original Eilenberg-MacLane paper). I ...
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### Colored arrangements of circles on the two sphere

Let me define a degree $n$ colored arrangement of circles on $S^2$ to be a collection $\mathcal{C}$ of $n$ disjoint, smoothly embedded circles $C_1,\dotsc, C_n\subset S^2$ together with a ...
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### Klein Bottle exception to the Heawood Conjecture [duplicate]

Possible Duplicate: The Klein bottle and the Heawood Conjecture It is well known that the Heawood Conjecture states that the bound for the number of colours which are sufficient to colour a ...
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### Presentation of the pure Artin groups

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by W= < T_1, \dots, T_n | T_i^2=1, T_iT_jT_i \ldots = T_jT_iT_j \ldots, i \neq ...
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### Once differentiable, piecewise degree three polynomials on triangulated planar domains

Here is an easily described, but very difficult, problem that I (and a number of other people) really would like to see solved during our life times. The basic problem is to compute the dimension of ...
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### Hamiltonian cycles and fundamental groups

I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the Lovász conjecture for Cayley graphs, I started noticing ...
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### Poset fiber theorems under a special assumption on the poset map?!

Hey everyone, I am facing the following problem: Say that a (order-preserving) poset map $f:P\to Q$ has property $(\star)$ if for all $q_1,q_2\in Q$ with $q_1\leq q_2$ and every $p_2\in f^{-1}(q_2)$ ...
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### How to determine whether a map between posets is a fibration

Hey everyone, is there a good way to determine whether a map of (the topological realizations of) posets is a fibration without explicitely proving that it has the homotopy lifting property? Robert
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### Is there a reset sequence?

There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone ...
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### What is the right notion of self-dual (two-dimensional) percolation in R^4?

For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...
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### Symmetric colorings of regular tesselations

Given a regular tesselation, i.e. either a platonic solid (a tesselation of the sphere), the tesselation of the euclidean plane by squares or by regular hexagons, or a regular tesselation of the ...
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### To what extent is convexity a local property?

A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
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### Does a triangulation without fixed simplex property always exist?

Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex ...
Suppose I have two $n$-sided polygons A and B. Is there a non-trivial upper bound on the number of parameters (eg. area, perimeter, etc) of the two polygons, that need to be the same, for A and B to ...