Related papers: The planar pure braid group is a diagram group

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first deals with connected…

We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n bodies all follow one another at regular intervals along…

In this paper we define a new family of groups which generalize the {\it classical braid groups on} $\C $. We denote this family by $\{B_n^m\}_{n \ge m+1}$ where $n,m \in \N$. The family $\{ B_n^1 \}_{n \in \N}$ is the set of classical…

Given a graph $\Gamma$ and a number $n$, the associated $n^{th}$ graph braid group $B_n(\Gamma)$ is the fundamental group of the unordered configuration space of $n$ points on $\Gamma$. \'{S}wi\k{a}tkowski showed that for a given $\Gamma$…

Let $\Sigma_g$ denote the closed orientable surface of genus $g$ and fix an arbitrary simplicial triangulation of $\Sigma_g$. We construct and study a natural surjective group homomorphism from the surface braid group on $n$ strands on…

A tanglegram consists of two rooted binary trees and a perfect matching between their leaves, and a planar tanglegram is one that admits a layout with no crossings. We show that the problem of generating planar tanglegrams uniformly at…

Let B_n(RP^2)$ (respectively P_n(RP^2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP^2. In this paper we study these braid groups, in particular the associated pure braid group short…

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no…

A regular bipartite graph $\Gamma$ is called semisymmetric if its full automorphism group $\mathrm{Aut}(\Gamma)$ acts transitively on the edge set but not on the vertex set. For a subgroup $G$ of $\mathrm{Aut}(\Gamma)$ that stabilizes the…

S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group…

Consider the unit ball, $B = D \times [0,1]$, containing $n$ unknotted arcs $a_1, a_2, ..., a_n$ such that the boundary of each $a_i$ lies in $D \times \{0\}$. The Hilden (or Wicket) group is the mapping class group of $B$ fixing the arcs…

If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We…

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, $4$-valent one-regular graphs of order $5p^2$, where $p$ is a prime, are classified

A graph $G$ is called matching covered if all of its edges are contained in some perfect matching of $G$. Furthermore, a cycle $C \subseteq G$ is called conformal if $G - V(C)$ has a perfect matching and $G$ itself is called cycle-conformal…

The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for…

Define a Garside monoid to be a cancellative monoid where right and left lcm's exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garside monoid. The family of Garside groups…

This paper gives a new, simplified presentation of the classical pure braid group. The generators are given by the squares of the longest elements over connected subgraphs, and we prove that the only relations are either commutators or…