If Γ is any finite graph, then the unlabelled configuration space of n points on Γ, denoted U C n Γ, is the space of n-element subsets of Γ. The braid group of Γ on n strands is the fundamental group of U C n Γ.We apply a discrete version of Morse theory to these U C n Γ, for any n and any Γ, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space U C n Γ strong deformation retracts onto a CW complex of dimension at most k , where k is the number of vertices in Γ of degree at least 3 (and k is thus independent of n).
Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration space $UC^n \Gamma$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $B_n\Gamma$, is the fundamental group of $UC^n \Gamma$. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" to get a partial description of the cohomology rings $H^*(B_n T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a large number of counterexamples to Ghrist's conjecture that braid groups of planar graphs are right-angled Artin groups.Comment: 25 pages, 7 figures. Revised version, accepted by the Journal of Pure and Applied Algebr
Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.Comment: 13 pages, final version, appears in Discrete Mathematics 201
Abstract. Let Γ be a graph. The (unlabeled) configuration space U C n Γ of n points on Γ is the space of n-element subsets of Γ. The n-strand braid group of Γ, denoted BnΓ, is the fundamental group of U C n Γ.This paper extends the methods and results of [11]. Here we compute presentations for BnΓ, where n is an arbitrary natural number and Γ is an arbitrary finite connected graph. Particular attention is paid to the case n = 2, and many examples are given.
Abstract. We study the geodesic growth series of the braid group on three strands, B3 := a, b|aba = bab . We show that the set of geodesics of B3 with respect to the generating set S := {a, b} ±1 is a regular language, and we provide an explicit computation of the geodesic growth series with respect to this set of generators. In the process, we give a necessary and sufficient condition for a freely reduced word w ∈ S * to be geodesic in B3 with respect to S. Also, we show that the translation length with respect to S of any element in B3 is an integer.
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