Paolo Bellingeri, Arthur Soulié
In this note, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.
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On Universal Virtual and Welded Braid Groups and Their Linear Representations
本文研究了通用虚拟辫群的线性表示，并探讨了其焊接辫群的结构与性质，包括其交换化、完美交换子群及有限商群等特征。
Let $WB_n$ be the welded (or loop) braid group on n strands, $n \geq 3$. We investigate commutator subgroup of $WB_n$. We prove that the commutator subgroup $WB_n'$ is finitely generated and Hopfian. We show that $WB_n'$ is perfect if and only if $n \geq 5$. We also compute finite presentation for $FWB_n'$, the commutator subgroup of the flat welded braid group $FWB_n$. Along the way, we investigate adorability of these groups.
Milnor invariants of braids and welded braids up to homotopy
本文研究同伦下纯焊接辫子群的结构，通过将其商群分解为迭代半直积解决Andreadakis问题并给出生成元表示，同时关联Milnor不变量并讨论经典辫子的对应情况。
Multi-welded twin groups
本文引入多焊接孪生群 $M_kWT_n$，证明其作为通用焊接辫群 $UW_n(k)$ 的商群，并研究其结构性质、交换化、换位子群完美性及表示理论。
Irreducible representations of welded braid group
本文研究焊接辫子群 $WB_n$ 的不可约表示，证明其在特定维数范围内不存在不可约表示，并对辫子群 $B_n$ 的若干不可约表示到 $WB_n$ 的扩张给出完整分类，同时发现新的不可约表示族。
Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitely generated if and only if $n \geq 4$, and $WB_n'$ is finitely generated for $n \geq 3$. Also we prove that $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB…
Using totally symmetric sets, Chudnovsky, Kordek, Li, and Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups, and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.
In the present paper the representation of the virtual braid group $VB_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that these already known representations are not faithful for $n \geq 4$ is verified. Using representations of $VB_n$, the virtual link group is defined. Also representations of welded braid group $WB_n$ are constructed and the welded link group is defined.
Characteristic subgroups and the R$_\infty$-property for virtual braid groups
本文证明了虚拟纯辫群、焊接纯辫群等在特定阶数下为相应辫群的特征子群，并证明虚拟辫群、无限制虚拟辫群等具有R$_\infty$性质，同时指出部分纯辫群在特定阶数下的该性质仍属未知。
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group.
Virtual braid groups, virtual twin groups and crystallographic groups
本文证明虚拟辫子群与虚拟孪生群模去对应纯群换位子群后的商为晶体群，刻画其有限阶元素与共轭类，并实现 Bieberbach 群及无限虚拟循环群。
In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid groups $vP_n$, as well as their `upper' variants, $wP_n^+$ and $vP_n^+$. We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.
Multi-virtual braid groups
本文研究了多虚拟辫群及其相关结构，介绍了多虚拟纯辫群、半纯辫群以及多焊合和多非限制辫群，并探讨了三股2虚拟辫群的结构及未来研究方向。
On virtual singular braid groups
本文研究了虚拟奇异辫群的代数性质，引入了基于单词指数和的数值不变量，确定了到对称群的同态，并给出了核的半直积分解。
In the paper, we construct a representation $θ:FVB_n\to{\rm Aut}(F_{2n})$ of the flat virtual braid group $FVB_n$ on $n$ strands by automorphisms of the free group $F_{2n}$ with $2n$ generators which does not preserve the forbidden relations in the flat virtual braid group. This representation gives a positive answer to the problem formulated by V. Bardakov in the list of unsolved problems in virtual knot theory and combinatorial knot theory by R. Fenn, D. Ilyutko, L. Kauffman and V. Manturov. Using this representation we construct a new group invariant for flat welded links. Also we find the …
In this note, we show that the sets of all stable commutator lengths in the braided Ptolemy-Thompson groups are equal to non-negative rational numbers.
In a recent paper by L. A. Bokut, V. V. Chaynikov and K. P. Shum in 2007, Braid group $B_n$ is represented by Artin-Burau's relations. For such a representation, it is told that all other compositions can be checked in the same way. In this note, we support this claim and check all compositions.
We give a simple characterization of braids that can be unplaited keeping separately their upper ends and their lower ends tied together
We view the $\tilde{A}$-type affine braid group as a subgroup of the $B$-type braid group. We show that the $\tilde{A}$-type affine braid group surjects onto the $A$-type braid group and we detect the kernel of this surjection using Schreier's Theorem. We then describe an injection of the $B$-type braid group into the $A$-type braid group which allows us finally to give a definition of affine links, as closures of affine braids viewed as A-type braids after composing the above injections, and we prove that the two conditions of Markov are necessary and sufficient to get the same affine closure…
We give a 3-page description of the Gassner invariant / representation of braids / pure braids, along with a description and a proof of its unitarity property.
Proof of Shvartsman's conjecture on braid groups of projective complex reflection groups
本文证明了关于复射影反射群的辫群猜想，并修正了Broué等人关于射影反射群的结果。
Khovanov-Seidel braid representation
本文为CIRM讲座讲义，通俗介绍Khovanov-Seidel对Artin-Tits群的范畴表示，强调其可计算性，并提及在几何群论中的潜在应用。
When the lower central series stops: a comprehensive study for braid groups and their relatives
本文研究群的下中心列何时停止的问题，引入多种工具并应用于辫群及其相关群，除射影平面上的一族分割辫群外均给出答案，部分情形完整计算了下中心列。
In this short note we provide an alternative proof of a theorem of Kapovich and Millson about the Malcev completion of an arbitrary Artin group, and determine the Malcev completion of the braid group of an irreducible finite complex reflection group.
Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. The goal of this review article is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. We also use it as an opportuni…
Low-dimensional indecomposable representations of the braid group $B_3$
本研究完成了特征0代数闭域上2维与3维辫群$B_3$可约但不可分解表示的等价类完整分类，并举例说明其应用价值。
A Note on Pure Braids and Link Concordance
该研究聚焦弦链协变群的代数性质，证明n股弦链协变群模掉其纯辫子群的商群仍为非阿贝尔群，推进了链环协变领域的群结构研究。
In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $ϕ\in Aut (B_n)$, decides whether $v=(ϕ(x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem.
The group of basis-conjugating automorphisms of the free group of rank $n$, also known as the McCool group or the welded braid group $PΣ_n$, contains a much-studied subgroup, called the upper McCool group $PΣ_n^+$. Starting from the cohomology ring of $PΣ_n^+$, we find, by means of a Gröbner basis computation, a simple presentation for the infinitesimal Alexander invariant of this group, from which we determine the resonance varieties and the Chen ranks of the upper McCool groups. These computations reveal that, unlike for the pure braid group $P_n$ and the full McCool group $PΣ_n$, the Chen r…
The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(\mathbb{R})$ of $n$ labelled points in $\mathbb{R}$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.
These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin's presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. …
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this monodromy representation using standard tools from the theory of hyperplane arrangements. In particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but that it cannot b…
A note on the pure cactus group of degree three and the configuration space of four points on the circle
本文明确描述了三次纯仙人掌群与圆上四点构型空间之间的关系。
This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literat…
Products of free groups inside graph braid groups
该研究针对图辫群的子群结构问题，在Świątkowski的自由阿贝尔子群结论基础上，证明可通过添加额外点得到同上同调维数的非阿贝尔自由群直积子群，并给出相关应用。
In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and $R$-matrix introd…
A note on the non-abelian tensor squares of linear groups
本文证明了有限生成线性群的非阿贝尔张量平方可能非线性，而辫群 $B_n$（$n>3$）的张量平方是线性的。
In this paper, we introduce a broad family of group homomorphisms that we name the Gauss-Epple homomorphisms. In the setting of braid groups, the Gauss-Epple invariant was originally defined by Epple based on a note of Gauss as an action of the braid group $B_n$ on the set $\{1, \dots, n\}\times\mathbb{Z}$; we prove that it is well-defined. We consider the associated group homomorphism from $B_n$ to the symmetric group $\text{Sym}(\{1, \dots, n\}\times\mathbb{Z})$. We prove that this homomorphism factors through $\mathbb{Z}^n\rtimes S_n$ (in fact, its image is an order 2 subgroup of the previo…
The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_\infty$-property and are not co-Hopfian for $n \ge 3$.
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