相关论文: From braid groups to mapping class groups

A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their…

The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi's…

For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of…

We give a method for constructing an interactive art piece which illustrates two different definitions of the braid groups, along with their faithful action on the free group. The box also demonstrates how all motions of points in the plane…

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus $g$ with $b$ boundary components and $n$ punctures, we show that these groups are isomorphic to some groups related to the braid groups and the…

Garside families have recently emerged as a relevant context for extending results involving Garside monoids and groups, which themselves extend the classical theory of (generalized) braid groups. Here we establish various characterizations…

Pure braid groups and pure mapping class groups of a punctured sphere have many features in common. In the paper the graded Lie algebra of the descending central series of the pure mapping class of a sphere is studied. A simple presentation…

The aim of this paper is to survey some aspects of mapping class groups with focus on their finite dimensional representations arising in topological quantum field theory.

We study a certain type of braid closure which resembles the plat closure but has certain advantages; for example, it maps pure braids to knots. The main results of this note are a Markov-type theorem and a description of how Vassiliev…

The present paper are the notes of a mini-course addressed mainly to non-experts. It purpose it to provide a first approach to the theory of mapping class groups of non-orientable surfaces.

Statistical and probabilistic characteristics of locally free group with growing number of generators are defined and their application to statistics of braid groups is given.

We define an action of Artin's braid group on a finite dimensional algebra.

We construct finite-dimensional projective representations of the mapping class groups of compact connected oriented surfaces having one boundary component using stated skein algebras.

This is a survey paper based on my talk at the Workshop on Orbifolds and String Theory, the goal of which was to explain the role of groupoids and their classifying spaces as a foundation for the theory of orbifolds.

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the…

We investigate braid group representations associated with unitary braided vector spaces, focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite…

A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods,…

The loop braid group is the motion group of unknotted oriented circles in $\mathbb{R}^3$. In this paper, we study their representations through the approach inspired by two dimensional topological phases of matter. In principle, the motion…

We describe the lower algebraic $K$-theory of the integral group ring of both the pure and full braid groups of the real projective plane $\mathbb{R}P^2$ with $3$ strings, as well as that of the integral group ring of the mapping class…

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.