Related papers: Diagrammatics for Coxeter groups and their braid g…
We start from any small strict monoidal braided Ab-category and extend it to a monoidal nonstrict braided Ab-category which contains braided bialgebras. The objects of the original category turn out to be modules for these bialgebras
We classify fully commutative elements in the affine Coxeter group of type $\tilde{A_{n}}$. We give a normal form for such elements, then we propose an application of this normal form: we lift these fully commutative elements to the affine…
We give the definition of presentations of linear monoidal categories. Our main result is that given a presentation of a linear monoidal category, we can produce a presentation of the same category as a linear category. We apply this result…
We introduce a monoidal category whose morphisms are finite partial orders, with chosen minimal and maximal elements as source and target respectively. After recalling the notion of presentation of a monoidal category by the means of…
We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on…
Given a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and an $\mathbb{E}_{\infty}$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{Fun}(M,…
We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way…
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure…
We provide conditions on the defining graph of a right-angled Coxeter group presentation that guarantees the boundary of any CAT(0) space on which the group acts geometrically will be locally connected. This is a revised version of a…
We introduce the notions of a half-balanced braided monoidal category and of its contraction. These notions give rise to an explicit description of the action of the Galois group of QQ on Teichmueller groupoids in genus 0, equivalent to…
We characterize folding patterns, the combinatorial options of folding minimal alcove-to-alcove galleries in affine Coxeter complexes positively with respect to Weyl chamber orientations of the Coxeter complex, by drawing a connection to…
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed…
We obtain a novel formula for characteristic polynomials of deformations of the Braid arrangement using the notion of levels of regions. As an application, we recover and strengthen results of Chen et al. on the characteristic polynomial of…
We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our…
We show the uniqueness and existence of the Euler form for a simply-laced generalized root system. This enables us to show that the Coxeter element for a simply-laced generalized root system is admissible in the sense of R.~W.~Carter. As an…
Hughes defined a class of groups that act as local similarities on compact ultrametric spaces. Guba and Sapir had previously defined braided diagram groups over semigroup presentations. The two classes of groups share some common…
We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise…
We define and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of…
In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy…
We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly…