Related papers: Virtual Braids and Cluster Algebras

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group $U_qsl_2$ at root of unity. In this paper,…

We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index…

We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the…

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change,…

We show that any of the new knot invariants obtained from Chern-Simons theory based on an arbitrary non-abelian gauge group do not distinguish isotopically inequivalent mutant knots and links. In an attempt to distinguish these knots and…

There is a well known injective homomorphism $\phi:{\mathcal {B}}_n \rightarrow {\rm Aut}(F_n)$ from the classical braid group ${\mathcal {B}}_n$ into the automorphism group of the free group $F_n$, first described by Artin. This…

Non-trivial braid-group representations appear as non-Abelian quantum statistics of emergent Majorana zero modes in one and two-dimensional topological superconductors. Here, we generate such representations with topologically protected…

This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M \setminus K$. We equip $M \setminus K$ with a hyperbolic metric $h$…

This article is dedicated to the computation of an explicit presentation of some asymptotically rigid mapping class groups, namely the braided Higman-Thompson groups. To do so, we use the action of these groups on the spine complex, a…

The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and…

We establish an identification between the spaces of $\alpha$-fusion trees in non-semisimple topological quantum computation (NSS TQC) and a family of homological representations of the braid group known as the Lawrence representations…

We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.

Motivated by an experimental study of groups generated by reflections in planar patterns of tangent circles, we describe some methods for constructing and studying representation spaces of holonomy groups of infinite volume hyperbolic…

An extension of the Artin Braid Group with new operators that generate double and triple intersections is considered. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple…

Let $UVB_n$ and $UVP_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups $UVB_n$ and $UVP_n$, and our main results are as follows: for $n\geq 5$, we give a…

A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found.…

Burau representation of the Artin braid group remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for…

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a braided tensor category, we give a detailed construction of the…

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…

We first motivate the study of a certain quotient of the loop braid category, both for the mathematics underpinning recent approaches to topological quantum computation; and as a key example in non-semisimple higher representation theory.…