Related papers: Topological complexity of spatial polygon spaces
Heron's formula states that the area $K$ of a triangle with sides $a$, $b$, and $c$ is given by $$ K=\sqrt {s(s-a) (s-b) (s-c)} $$ where $s$ is the semiperimeter $(a+b+c)/2$. Brahmagupta, Robbins, Roskies, and Maley generalized this formula…
By a formula of Farber the topological complexity TC(X) of a (p-1)-connected, m-dimensional CW-complex X is bounded above by (2m+1)/p+1. There are also various lower estimates for TC(X) such as the nilpotency of the ring $H^*(X\times…
Consider the space of `long knots' in R^n, K_{n,1}. This is the space of knots as studied by V. Vassiliev. Based on previous work of the authors, it follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson algebra. A…
We describe the set of possible vector valued side lengths of n-gons in thick Euclidean buildings of rank 2. This set is determined by a finite set of homogeneous linear inequalities, which we call the generalized triangle inequalities.…
Consider the configuration spaces of manifolds. We give a precise formula for the integral cohomological dimension (the degree of top non-trivial integral cohomology group) of unordered configuration spaces of manifolds with non-trivial…
Regular polygonal complexes in euclidean 3-space are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The…
We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of…
We continue our study of the topology of the spaces of $m$ tuples of real polynomials with common degree $d$ and without common roots of multiplicity $n$, and in particular their stability properties with respect to $d$. In an earlier paper…
Circuit complexity for two-dimensional topological quantum field theories (2D TQFT) was defined by Couch, Fan, and Shashi in [12]. In this paper, we study complexity for the 2D TQFT given by quantum cohomology of compact symplectic…
A class of topological spaces is topologically rigid if any two spaces with the same fundamental group are also homeomorphic. Topological rigidity, in addition to its intrinsic interest, has been useful for solving abstract commensurability…
This is a short survey of Riemannian geometric applications of Lp-cohomology of thick spaces, p not equal to 2.
We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$…
In this paper, we introduce the n-th discrete topological complexity and study its properties such as its relation with simplicial Lusternik-Schnirelmann category and how the higher dimensions of discrete topological complexity relate with…
The triangulation complexity of a compact 3-manifold is the minimal number of tetrahedra in any triangulation of the 3-manifold. We compute the triangulation complexity of all elliptic 3-manifolds and all sol 3-manifolds, to within a…
We study the topology of the complex points of the algebraic loop space of a smooth curve.
Following Ghomi and Tabachnikov we study topological obstructions to totally skew embeddings of a smooth manifold M in Euclidean spaces. This problem is naturally related to the question of estimating the geometric dimension of the stable…
Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for…
These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…
We prove the theorem mentioned in the title, for ${\mathbb{R}}^n$, where $n \ge 3$. The case of the simplex was known previously. Also, the case $n=2$ was settled, but there the infimum was some well-defined function of the side lengths. We…
The paper establishes that the rank of a regular polygonal complex in 3-space E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in 3-space are the eight regular 4-apeirotopes.