Questions tagged [braid-groups]

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8
votes
1answer
177 views

Geometric intuition behind Garside's paper?

I apologize in advance for a somewhat wishy-washy question. I just read the paper "The Braid Group and Other Groups" by F. A. Garside in which he solves the conjugacy problem for the braid ...
3
votes
1answer
129 views

From braid representations to link invariants

If one has a $\mathbb{C}$-linear representation of the braid algebra into e.g. the Temperley-Lieb algebra i.e. $\rho:\mathbb{C}[B_{n}]\to TL_{n}(\delta)$, we can deduce a skein relation $\mathcal{S}$....
0
votes
1answer
66 views

Fundamental group to groupoid : bijection between homotopy classes?

I'm looking at the fundamental group $\pi_{1}(M)$ of the $n^{th}$ unordered configuration space $M$ of $\mathbb{R}^{d}$. In particular, it's well-known that $\pi_{1}(M)\cong S_{n}$ (symmetric group) ...
3
votes
0answers
131 views

Braids of fibered knots

There are some theorems saying that the closure of a braid of a specific form is fibered. For instance, a theorem of Stallings says that the closure of a homogeneous braid is a fibered knot. I am ...
9
votes
2answers
387 views

Braid groups and Kazhdan's property (T)

In Nica's dissertation Group actions on median spaces, we can read the following assertion: Braid groups do not contain infinite subgroups satisfying Kazhdan's property (T). This is used in order to ...
9
votes
0answers
103 views

Artin groups of type $D_n$ as mapping class groups?

According to Allcock (Braid Pictures for Artin groups, https://arxiv.org/abs/math/9907194), the Artin group $A(D_n$) of type $D_n$ may be realized as an index 2 subgroup of the orbifold fundamental ...
1
vote
0answers
39 views

How to get the braid word of the tubes in a reducible braid?

If a braid is reducible, it means that the strands can be separated into groups of strands (less groups than number of strands), such that each group is running inside one tube. The tubes themselves ...
3
votes
1answer
62 views

Knots with a braid presentation with only positive or negative crossings on each fixed position

I am interested in the following class of knots $K$: $\{$$K$ has a braid presentation such that for any fixed position $k$, either only positive or negative powers of $\sigma_k$ appear in the braid ...
12
votes
2answers
632 views

Several questions about Gauss's mathematical conception of braids

I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits ...
5
votes
1answer
168 views

Integral homology of braid groups as a ring

Let $Br_k$ denote the braid group on $k$ strands. In Corollary A.4 of "Homology of Iterated Loop Spaces" (Page 348), Cohen-Lada-May compute $H_i(Br_k;\mathbb Z)$ as an abelian group for each ...
6
votes
2answers
207 views

CW-presentation of configurations of points in plane and space

I know from the the theory of Artin groups that, as the $K(\pi,1)$ conjecture is known for Braids group, that using Salvetti complexes we have a fairly explicit finite CW-complex presentation of the ...
2
votes
1answer
181 views

Linear representations of Hurwitz braid group with small degree

In Bigelow: Does the Jones Polynomial detect the Unknot?, J. Knot Theory Ramifications, 11, 493-505 (2002), Corollary 6.2. ff., a non-trivial braid $\beta$ in the kernel of the specialized Burau ...
1
vote
0answers
79 views

When is the action of the braid group on tensor powers of Yetter-Drinfeld modules faithful?

Let $V$ be a Yetter-Drinfeld module over a Hopf algebra $H$ with invertible antipode. Recall that $V$ is a braided vector space with braiding $\Psi\colon V\otimes V\to V\otimes V, v\otimes w\mapsto v_{...
2
votes
0answers
64 views

On $\Psi$-generating paths in the Bruhat order of a Weyl group

Let $W$ be a Weyl group with roots $R$ and positive roots $R^+$. Let $v\in W$ of length $r$. We call $\mathbb{m}=(\alpha_1,\ldots,\alpha_r)\in(R^+)^r$ a Bruhat path from $1$ to $v$ if $1\lessdot s_{\...
10
votes
0answers
222 views

Surjective homomorphisms between braid groups

There cannot be a surjective homomorphism $B_2 \to B_n$ for any $n > 2$ because $B_2$ is commutative and $B_n$ is not. It seems plausible that if $m < n$, there cannot be a surjective ...
3
votes
1answer
100 views

Cohomology of the moduli space of rational curves with $n$ marked points with spin structure

Consider $\mathcal{M}_{0,n}$- the moduli space of rational curves with $n$ marked points. A map $$ p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z}) $$ ...
5
votes
1answer
208 views

Do there exist any variational principles on the space of braids (or knots)?

This is very speculative question and I do not know where to start looking up the literature, or if what I am looking for is even mathematically possible/meaningful. Q: I am interested in finding out ...
3
votes
1answer
275 views

Permuting $n$ points in a $2$-manifold

Given $n$ points on a connected $2$-manifold $M$, I'd like to consider the homotopy classes of paths that "permute" these points. Edit (Clarifying what I mean by this): Given a set of $n$ ...
2
votes
0answers
96 views

Software for finding conjugates in the Braid group

The conjugacy problem for the braid group was solved by Garside, and gives an algorithm for determining whether two braids are conjugate. Since this algorithm is rather tedious, I was wondering if ...
9
votes
0answers
260 views

Finite quotients of surface braid groups

Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$. There is a braid $A_{12} \in \Sigma_b$ ...
8
votes
5answers
810 views

When do elements in the braid group $B_n$ commute?

I have been looking around for an answer to this question, but I have not been able to find anything. My question is: Is it known how to tell whether two elements $b_1, b_2 \in B_n$ commute? EDIT: ...
6
votes
0answers
109 views

Is there a natural, purely group-theoretic definition of the virtual braid group?

The braid group $B_n$ has the well-known presentation $$\left<\sigma_i,i=1\ldots n-1\, \left| \begin{cases}\sigma_i\sigma_j=\sigma_j\sigma_i & |i-j|>1\\\sigma_i\sigma_j\sigma_i=\sigma_j\...
2
votes
2answers
264 views

Algorithm for identifying reducible braids

If $\vec{n} = (n_1,...,n_k)$ is a vector of integers, there seems to be a well-defined homomorphism $B_k \ltimes \left(B_{n_1} \times \cdots \times B_{n_k}\right) \to B_N$ where $N = \sum n_i$ and $...
2
votes
0answers
85 views

Cohomology of colored braid groupoids

Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, ...
3
votes
0answers
559 views

Braided lobsters

If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define ...
2
votes
0answers
102 views

Is the action of free self-distributive algebras on racks computable in polynomial time?

Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the mapping where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Then $B_{\...
2
votes
0answers
99 views

Does shifted conjugacy still give you free self-distributive algebras on one generator for quotient groups of the braid groups?

Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the group homomorphism where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ for all $i>0$. ...
3
votes
0answers
88 views

What are the composition series for these series of groups?

A rack is an algebra $(X,*,*^{-1})$ that satisfies the identities $x*(y*z)=(x*y)*(x*z)$ and $x*(x*^{-1}y)=x*^{-1}(x*y)=y$. If $X$ is a rack then define a homomorphism $\phi_{n,X}:B_{n}\rightarrow \...
2
votes
0answers
203 views

Does the Burau representation of braids distinguish between distinct elements of the free self-distributive algebras on one generator?

A well-known but now mostly solved problem in group theory is the question of whether the Burau representation of the braid groups is faithful. It turns out that this representation is not faithful ...
5
votes
0answers
135 views

Do the ternary braid groups arise in algebraic topology?

Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$ and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|...
5
votes
1answer
193 views

Examples of Yang-Baxter monoids

Then we say that an algebra $(X,f,g,\circ,1)$ is a Yang-Baxter monoid if it satisfies the following identities: $(X,\circ,1)$ is a monoid, $f(x,1)=1,f(1,x)=x,g(x,1)=x,g(1,x)=1$ $x\circ y=f(x,y)\circ ...
5
votes
0answers
88 views

Permutative Yang-Baxter monoids

Suppose that $f,g:X^{2}\rightarrow X,T:X^{2}\rightarrow X^{2}$ are mappings such that $T(x,y)=(f(x,y),g(x,y))$. An element $1\in X$ is said to be an identity if $T(1,x)=(x,1),T(x,1)=(1,x)$. The ...
3
votes
0answers
91 views

Is the variety of algebras that satisfy the Yang-Baxter equation generated by its finite members?

Suppose that $f,g:X^{2}\rightarrow X$, and $T:X^{2}\rightarrow X^{2}$ is the function where $T(x,y)=(f(x,y),g(x,y))$. Then $(X,f,g)$ is said to satisfy the Yang-Baxter equation if $(T\times 1_{X})\...
4
votes
0answers
129 views

Does the Hurwitz action of the braid group on rank-into-rank embeddings tend to increase the critical points?

An algebraic structure $(X,*)$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$. Suppose that $X$ is a self-distributive algebra. Then the positive braid monoid $B_{...
4
votes
2answers
297 views

Is the *unreduced* Burau representation unitary?

In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ ...
4
votes
0answers
103 views

semisimplicity of maps in braided vector spaces

Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$. This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\...
2
votes
1answer
167 views

Basis for Annular Skein Algebra

Background/Notation: Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis $\{T_{w}\}_{w\in S_{...
6
votes
1answer
135 views

Ordinary differential operators satisfying braid relation?

Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...
12
votes
1answer
446 views

Formality of the 2nd ordered configuration space of a closed Riemann surface

If $X$ is a smooth manifold, we define its kth ordered configuration space as $$F_kX:=\{(x_1, \ldots,x_k) \; | \; x_i \neq x_j \,\, \mathrm{if} \, \, i \neq j\},$$ in other words, $F_kX = X^k - \Delta,...
6
votes
2answers
285 views

Epimorphisms from the genus $2$ surface braid group to finite groups

This question is somehow related to my previous MO question Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$; for the reader convenience, let me write down again the relevant ...
9
votes
0answers
148 views

Interactions between pseudoline arrangements and braid groups?

It is common to represent pseudoline arrangements as wiring diagrams:                     Fig. from: "Hamiltonicity and colorings of arrangement ...
7
votes
1answer
238 views

Homotopy type of the semi-simplicial set of symmetric groups

Consider the collection of symmetric groups $\{\Sigma_n\}_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma_{...
3
votes
1answer
320 views

Centralizer of a generator in a braid group

Given a braid group $$ B_n \simeq \left\langle x_1,\ldots,x_{n-1} \middle| \begin{array}{l} x_ix_j = x_jx_i, \;\text{for } |i-j|>1 \\ x_ix_{i+1}x_i = x_{i+1}x_ix_{i+1} \end{array} \right\rangle $$...
5
votes
0answers
107 views

Question about terminology, and reference request related to the braid operad

Let $\Delta_n$ stand for the Garside element of the braid group $B_n$. It turns out that the family of all Garside elements have the following ``operadic'' property: $$ \Delta_n\left[ \Delta_{k_1},\...
8
votes
1answer
258 views

Recovering information about braids from their decomposition into positive and negative braids

Suppose that $b$ is a braid. Then $b$ can be uniquely written as $D_{RL}(b)^{-1}N_{RL}(b)$ where $D_{RL}(b),N_{RL}(b)$ are the unique positive braids such that $b=D_{RL}(b)^{-1}N_{RL}(b)$ and where $...
21
votes
1answer
591 views

What is the cohomological dimension of the commutator subgroup of the pure braid group?

I'm interested in computing the cohomological dimension of the commutator subgroup $[P_n,P_n]$ of the pure braid group $P_n$. I wasn't able to find a reference in the literature. Because $[P_n,P_n]$ ...
17
votes
1answer
501 views

Cohomology of braid groups with coefficients in the group ring

Let $\mathbf B_n$ be the braid group on $n$ strings. What is known about the cohomology of $\mathbf B_n$ with coefficients in its integral group ring: $H^*(\mathbf B_n;\mathbb Z \mathbf B_n)$?
5
votes
1answer
435 views

The action of the mapping class group of a punctured disk on the boundary at infinity of the universal cover

Let $\mathbb{D}\subset\mathbb{C}$ be the unit disk, and remove $n\geq 2$ of its points $P$. The resulting object will be called the punctured disk $\mathbb{D}_n$ in the following. I am interested in ...
6
votes
0answers
105 views

Braid groups representations on infinite dimensional vector spaces

Let $V$ be an infinite dimensional complex vector space. Let $R:V\otimes V\to V\otimes V$ be a solution to the quantum Yang Baxter Equation. In other words: $R$ is invertible and satisfies the ...
6
votes
2answers
230 views

Braid groups on topological spaces

The configuration space $C_n(M)$ of $n$ particles in some connected graph $M$ (thought of as the topological realisation of a one-dimensional CW-complex) is $$M^n \backslash \{ (x_1, \ldots, x_n) \...