Braid Group Research Articles

Minimal Generating Sets and the Abelianization for the Quasitoric Braid Group

Minimal Generating Sets and the Abelianization for the Quasitoric Braid Group

Quotients of braid groups by their congruence subgroups

The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation B n → G L n ( Z ) B_n \to GL_{n}(\mathbb {Z}) . We find the image of such congruence subgroups in G L n ( Z ) GL_{n}(\mathbb {Z}) —an open problem posed by Dan Margalit. Additionally, we characterize the quotients of braid groups by their congruence subgroups in terms of symplectic congruence subgroups.

The algebraic structure of hyperbolic graph braid groups

The algebraic structure of hyperbolic graph braid groups

Monodromy of stratified braid groups, II

The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group. We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} induced by the logarithmic derivative df/f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{d}f/f$$\\end{document} of a polynomial in the stratum.

The Borsuk–Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles

Let M and N be fiber bundles over the same base B, where M is endowed with a free involution τ over B. A homotopy class δ∈[M,N]B (over B) is said to have the Borsuk–Ulam property with respect to τ if for every fiber-preserving map f:M→N over B which represents δ there exists a point x∈M such that f(τ(x))=f(x). In the cases that B is a K(π,1)-space and the fibers of the projections M→B and N→B are K(π,1) closed surfaces SM and SN, respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map f:M→N over B has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of M, the orbit space of M by τ and a type of generalized braid groups of N that we call parametrized braid groups. As an application, we determine the homotopy classes of fiber-preserving self maps over S1 that satisfy the Borsuk-Ulam property, with respect to all involutions τ over S1, for the torus bundles over S1 with M=N=MA and A=[1n01].

Nielsen fixed point theory for split n-valued maps on the Klein bottle

In this work we begin the study of n-valued maps on the Klein bottle, denoted by K, where we focus on the split ones. We provide an explicit description of Pn(K) (the n-th pure braid group of K) as an iterated semi-direct product of the form Fn⋊θn−1(⋯⋊(F2⋊θ1π1(K))), where the Fi′s are free groups on i letters. Given a split n-valued map Φ={f1,…,fn} its pointed homotopy class is determined by a pair of braids in Pn(K). We also provide a formula for N(Φ), the Nielsen number of Φ, which is completely determined by two braids, which in turn also determine the homotopy classes of the functions fi′s. If Φ={f1,f2} is a 2-valued map with N(Φ)=0, we show that there exists at least one 2-valued map Φ′={f1′,f2′}, such that Φ′ is fixed point free and for i=1,2 it holds that [fi]=[fi′], where [] denotes a pointed homotopy class of maps. Finally, we display an infinite family of pointed homotopy classes in [K,F2(K)], such that N(ϕ)=0, for any ϕ in the family. Furthermore, the map from [K,F2(K)] to [K,K×K], induced by the inclusion F2(K)→K×K, takes this family to one single element in [K,K×K]. We do not know if these 2-valued maps of the family can be deformed to fixed point free maps.

Mini-Worskhop: Artin Groups meet Triangulated Categories

Artin and Coxeter groups are naturally occurring generalisations of the braid and symmetric groups respectively. However, unlike for Coxeter groups, many basic group theoretic questions remain unanswered for general Artin groups – most notably the K(\pi,1) -conjecture for Artin groups remains open except for certain special families of Artin groups. Recently, Artin groups have also appeared as groups acting on triangulated categories, where the associated spaces of Bridgeland’s stability conditions provide new realisations of the corresponding K(\pi,1) spaces. The aim of the workshop is to bring together experts and early career researchers from two seemingly different areas of research: (i) geometric and combinatorial group theory and topology, and (ii) triangulated categories and stability conditions, to explore their intersection via the K(\pi,1) -conjecture.

Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group

AbstractWe first construct an action of the extended double affine braid group $$\mathcal {\ddot{B}}$$ B ¨ on the quantum toroidal algebra $$U_{q}(\mathfrak {g}_{\textrm{tor}})$$ U q ( g tor ) in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of $$\mathcal {\ddot{B}}$$ B ¨ we produce automorphisms and anti-involutions of $$U_{q}(\mathfrak {g}_{\textrm{tor}})$$ U q ( g tor ) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and $$k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$$ k 0 a 0 ⋯ k n a n up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type A due to Miki which have been instrumental in the study of the structure and representation theory of $$U_{q}(\mathfrak {sl}_{n+1,\textrm{tor}})$$ U q ( sl n + 1 , tor ) .

Robertson’s conjecture and universal finite generation in the homology of graph braid groups

We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfoot–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs—in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Proudfoot–Ramos, implying that the category of cographs is Noetherian, a result of potential independent interest.

Cluster realisations of ıquantum$\imath {\rm quantum}$ groups of type AI

AbstractThe group of type is a coideal subalgebra of the quantum group , associated with the symmetric pair . In this paper, we give a cluster realisation of the algebra . Under such a realisation, we give cluster interpretations of some fundamental constructions of , including braid group symmetries, the coideal structure and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of , which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct the Poincare‐Birkhoff‐Witt (PBW)‐bases for the integral form.

The Transport of Species of Structures along the Braid Group

The purpose of this paper is to present in an introductory was the notion of transport of t-structures of a given species. The letter t symbolizes a braid with m strands that is performed on each element of [U]m.

The action of [formula omitted]-shadows on child's drawings

GT-shadows [8] are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group GTˆ introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB4, that are normal in B4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of GTˆ. We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, GTˆ, and the absolute Galois group GQ of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid GTSh♡ of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over Q. Finally, we describe selected examples of non-Abelian child's drawings.

Spherical objects in dimensions two and three

This paper classifies spherical objects in various geometric settings in dimensions two and three, including both minimal and partial crepant resolutions of Kleinian singularities, as well as arbitrary flopping 3 -fold contractions with only Gorenstein terminal singularities. The main result is much more general: in each such setting, we prove that all objects in the associated null category \mathcal{C} with no negative Ext groups are the image, under the action of an appropriate braid or pure braid group, of some object in the heart of a bounded t-structure. The corollary is that all objects x which admit no negative Exts, and for which \mathrm{Hom}_{\mathcal{C}}(x,x)=\mathbb{C} , are the images of the simples. A variation on this argument goes further and classifies all bounded t-structures on \mathcal{C} . There are multiple geometric, topological and algebraic consequences, primarily to autoequivalences and stability conditions. Our main new technique also extends into representation theory, and we establish that in the derived category of a finite-dimensional algebra which is silting discrete, every object with no negative Ext groups lies in the heart of a bounded t-structure. As a consequence, every semibrick complex can be completed to a simple-minded collection.

Braiding Fibonacci anyons

Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document}n acting on n-point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n. Finally, we construct N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} qubit computational spaces in terms of conformal blocks of 2N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 2\\mathcal{N} $$\\end{document} + 2 Fibonacci anyons.

Braided quantum symmetries of graph C∗-algebras

We prove the existence of a universal braided compact quantum group acting on a graph [Formula: see text]-algebra in the category of [Formula: see text]-[Formula: see text]-algebras with a twisted monoidal structure, in the spirit of the seminal work of Wang. To achieve this, we construct a braided analogue of the free unitary quantum group and study its bosonization. As a concrete example, we compute this universal braided compact quantum group for the Cuntz algebra.

The centre of a finitely generated strongly verbally closed group is almost always pure

ABSTRACT The assertion in the title implies that many interesting groups (for example, all non-abelian braid groups or $\textbf{SL}_{100}(\mathbb{Z})$) are not strongly verbally closed, i. e., they embed into some finitely generated groups as verbally closed subgroups, which are not retracts.

Shi arrangements and low elements in Coxeter groups

AbstractGiven an arbitrary Coxeter system and a non‐negative integer , the ‐Shi arrangement of is a subarrangement of the Coxeter hyperplane arrangement of . The classical Shi arrangement () was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for . As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in and that the union of their inverses form a convex subset of the Coxeter complex. The set of ‐low elements in were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in . In this article, we generalize and extend Shi's results to any Coxeter system for any : (1) the set of minimal length elements of the regions in a ‐Shi arrangement is precisely the set of ‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.

Homomorphisms of commutator subgroups of braid groups with small number of strings

Homomorphisms of commutator subgroups of braid groups with small number of strings

Necessary and sufficient conditions for the irreducibility of a linear representation of the braid group Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_n$$\\end{document}

Valerij G. Bardakov and P. Bellingeri introduced a new linear representation ρ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\rho }_F$$\\end{document} of degree n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} of the braid group Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_n$$\\end{document}. We study the irreducibility of this representation. We prove that ρ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\rho }_F$$\\end{document} is reducible to the degree n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document}. Moreover, we give necessary and sufficient conditions for the irreducibility of the complex specialization of its n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document} degree composition factor ϕ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\phi }_F$$\\end{document}.

Mutations of numerically exceptional collections on surfaces

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.