Related papers: The level four braid group
We study Dehn--Seidel twists on configurations of Lagrangian spheres in symplectic $K3$ surfaces, using tools from Seiberg--Witten theory. In the case of $ADE$ configurations of Lagrangian spheres, we prove that a naturally associated…
We introduce and study a family of groups $\mathbf{BB}_n$, called the blocked-braid groups, which are quotients of Artin's braid groups $\mathbf{B}_n$, and have the corresponding symmetric groups $\Sigma_n$ as quotients. They are defined by…
We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the…
We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is…
We characterize unitary representations of braid groups $B_n$ of degree linear in $n$ and finite images of such representations of degree exponential in $n$.
This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some…
In this paper we determine the automorphism groups of the profinite braid groups with four or more strings in terms of the profinite Grothendieck-Teichm\"uller group.
We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in…
We explicitly describe unitary representations of mixed braid groups on the cohomology of Abelian branched covers of $\mathbf{CP}^1$ . We show that the image of the representation is generated by complex reflections and relate it to the…
We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with…
We describe the fundamental groups of ordered and unordered $k-$point sets in the n-dimensional complex space $C^n$ generating an affine subspace of fixed dimension.
We prove that braid group representations associated to braided fusion categories and mapping class group representations associated to modular fusion categories are always semisimple. The proof relies on the theory of extensions in…
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility,…
The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural…
We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.
Using the recoupling theory, we define a representation of the pure braid group and show that it is not trivial.
We consider a certain modification of the group $G^3_n$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G^3_n$. The braid representation induced is powerful enough to…
In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.
A class of subgroups is obtained for symmetric groups using signed Brauer diagrams.
Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory…