The braid-groups tag has no usage guidance.

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### Partially permutative matrices

Let $V$ be a finite dimensional vector space over a field $K$. Then a map
$L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\...

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votes

**1**answer

128 views

### Are there any unitary matrices which satisfy the Yang-Baxter equation which are universal for quantum computation?

Let $H$ be a finite dimensional hilbert space. Let $L:H\otimes H\rightarrow H\otimes H$ be a unitary transformation. Then the equation
$$(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\...

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59 views

### Do the normal forms for braid groups really conceal information about better than randomly applying the braid relations?

In braid based cryptography, one typically wants to conceal the way a certain braid $b$ has been obtained. One therefore puts $b$ into some normal form. Since every braid has a unique normal form, the ...

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votes

**1**answer

141 views

### Are braid groups conjugacy separable?

I would like to re-ask a question that was raised in the comments here:
Normal subgroups of braid groups
Namely, a group $G$ is called conjugacy separable, if it holds that for every two elements $x,...

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196 views

### Cohomology of a configuration space of points on $\mathbb C^\times$ with an additional restriction

Let $Conf_{1,n}^3$ be the configuration space of collections of $n$ distinct numbered points on the annulus $\mathbb C^\times$ with an imposed restriction: for any $r\in \mathbb R^+$ the circle $\...

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votes

**1**answer

89 views

### A three-rope tangle that is equal to its mirror tangle

In the same way that a figure-eight knot is equal (after a suitable rotation) to its mirror knot, I am looking for simple tangles made of three ropes that are equal to their mirror tangle.
The ...

**3**

votes

**1**answer

158 views

### Braid group: Can a left-twist increase the number of right twists?

Disclaimer: This question was first posted on math.se without any answer.
This is something that naturally occurs in my research, but I am no expert on this - it feels like a natural question so I am ...

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76 views

### How long does it take for the action of the braid monoids on Laver tables to become trivial?

Let $A_{n}$ denote the classical Laver table of cardinality $2^{n}$.
Let $B_{n}^{*}$ denote the positive (including the identity) braid monoid on $n$ elements generated by $\sigma_{1},...,\sigma_{n}$....

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vote

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80 views

### coefficient of homology of configuration spaces over real projective spaces

In the slides Characteristic Classes of Surface Bundles
and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology?
Could the coefficient be an ...

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vote

**0**answers

134 views

### the braid groups on the sphere

The braid group $B_n(S^2)$ on the sphere possesses a finite element of order which generates a cyclic group $M$ in $B_n(S^2)$. My question is this:
What is the index of $H$ in $B_n (S^2)$?
Same ...

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votes

**0**answers

55 views

### cohomolgy of braid groups with elements of finite order

We have to braid groups with elements of finite order: 1 braid groups over the 2-sphere and brai group over the projective plane
can you give me a more detail on the cohomology of those groups?

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vote

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142 views

### Braids with an infinite number of strings

Has anyone developed a theory for braids with an infinite number of strings?

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71 views

### Expected shortest word length\depth in Braid groups from set of all braids with length L

Consider the set of all positive braids on n strands, with a fixed length L.
$$B^+_{n,L}:=\{\beta\in B_n:\beta=\sigma_{i_1}\sigma_{i_2}\ldots\sigma_{i_L},1\leq i_k \leq n-1\}$$.
Using the relations,
...

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votes

**1**answer

197 views

### Generators of pure braid groups of arbitrary Coxeter groups

Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma_i\in A$. Let $P$ be the "pure braid group", the kernel of the ...

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vote

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99 views

### Homology of spherical braid groups

By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...

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132 views

### What's the story with the Hopf fibration and the Jacobi identity?

I like the Hopf fibration of the 3-sphere $S^3$ enough that I found a nice way to make a physical model of it. All you need is to combine a bunch of key rings in such a way that (ii) every pair of ...

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**1**answer

236 views

### $\pi_1$ of the moduli of G-bundles on elliptic curves and the double affine braid group

For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset ...

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votes

**2**answers

613 views

### Braid group on 4 strands

Consider the braid group $B_4$ with generators
$a=\ $,
$b=\ $
and $c=\ $
Assume
$$\alpha, \alpha'
\in
\langle a,c\rangle{\smallsetminus}(\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle c\...

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votes

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142 views

### A question of braid words

Let $(W,S)$ be a Coxeter group, let $B(W,S)$ be the corresponding braid or Artin-Tits group. Set $S=\{s_1,\dots, s_n\}$ and denote by $\bf{S}=\{\sigma_1,\dots, \sigma_n\}$ the corresponding generators ...

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votes

**1**answer

107 views

### unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then
$$
B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1},
$$
$$
B(S^n,2)\simeq \mathbb{R}P^n.
$$
Hence
$
(*)
$
$$
B(\...

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votes

**1**answer

148 views

### presentations of subalgebras

Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many ...

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votes

**1**answer

144 views

### Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...

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votes

**1**answer

171 views

### Braid relations $n_\alpha n_\beta n_\alpha \ldots = n_\beta n_\alpha n_\beta \ldots $ in arbitrary reductive groups

I'm currently trying to prove or disprove the following claim. First let me set up some notation.
Let $G$ be a connected reductive group over a field $K$, let $S \leq Z \leq N \leq G$ be respectively ...

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votes

**1**answer

168 views

### Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group?

Is the braid group with $n$ strings $\mathcal{B}_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$)
Or is it known that it cannot be such a lattice ?

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181 views

### unordered configuration space of pointed space

Let $(X,*)$ be a pointed topological space.
Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid \forall i\neq j: x_i\neq x_j, \}$.
Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.
Is there an ...

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votes

**0**answers

94 views

### Are Markov traces matrix traces?

When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere.
Then when the list "Questions that may already ...

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131 views

### How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation
$$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i σ_{i+...

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votes

**1**answer

121 views

### Generators of the colored braid group (two colors), reference

I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white.
It is easy to find a set of generators for $B_{n,n}$:
$$
\begin{cases}
\...

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votes

**2**answers

203 views

### How can i change 8_19 to (3,4)-torus knot K(3,4)?

In the knot table, it is well-known that 8_19 is (3,4)-torus knot. But, it is not clear to me. How can i change 8_19 to (3,4)-torus knot K(3,4)? Moreover, it is well-known that braids of two ...

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72 views

### Framed braids and local systems

Let me start by admitting that my question is going to be somewhat vague. But hopefully it is one of these vague questions that can be immediately answered by an expert in the appropriate area.
...

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**1**answer

201 views

### homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold.
Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.
Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered ...

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299 views

### Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups

I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper:
(The proof is not finished yet but I am very confused by now.)
...

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**1**answer

339 views

### Is there a “good” reason that the universal central extension of $SL(2,\mathbb Z)$ is $Br_3$?

Let
$$ Br_3 := \langle \tau_1,\tau_2\ :\ \tau_1 \tau_2 \tau_1 = \tau_2 \tau_1 \tau_2 \rangle $$
be the braid group on three strands, and consider the surjection
$$\phi : Br_3 \twoheadrightarrow SL_2(\...

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votes

**1**answer

148 views

### Lower Central Series of Pure Braid Groups?

What is the lower central series $\Gamma_k(P_n)$, where $P_n$ is the pure braid group with $n$ strands? We know that $P_n$ is generated by elements $A_{i,j}$; do we know the generators of $\Gamma_k(...

**6**

votes

**1**answer

270 views

### Action of the homotopy braid groups on reduced free groups

Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...

**0**

votes

**1**answer

210 views

### Action of the pure braid group on the commutator subgroup of a free group

Let $P=P_n$ be the pure braid group on $n$ strands and $F=F_n$ the free group on $n$ generators. I'm interested in a nice description of the action of $P$ on the derived subgroup $F'$ which somehow ...

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votes

**1**answer

121 views

### Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.
I am ...

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270 views

### Testing the faithfulness of group homomorphisms by testing on the level of induced Lie Algebras

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the
lower central series of $G$. For each $k\geq 1$, set
$\mathrm{gr}_k(G)=\Gamma_G(k)/\Gamma_G(k+1)$ and
$$\mathrm{gr}_*(G):=\...

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**1**answer

187 views

### (Alternative) Presentation for the pure braid group of the sphere

First I need some notation (it's all standard I think). For a manifold $M$, let $F_nM = F_{0,n}M$ be the space of $n$-tuples of distinct points on $M$ ; let $B_nM = B_{0,n}M = F_nM / \Sigma_n$. When $...

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317 views

### Are homotopy braid groups residually nilpotent?

A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid ...

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**1**answer

161 views

### Associated graded Lie algebra of braid groups

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and $$\mathcal{L}_G:=\bigoplus_{k\...

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143 views

### Braids, pure braids and Dehn twists

Consider the braid group with $n$ strands $B_n$. Each braid can be drawn (say) from bottom to top as $n$-intertwining strictly monotonic strands. Moreover, the group $B_n$ is generated by $n-1$ ...

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157 views

### 2-cells in the configuration space

My overarching question is why in the Artin presentation of the (geometrically defined) braid group there are no more than two relations. I've learned one way to prove this is using the fact that the ...

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302 views

### Presentation of Homotopy Pure Braid Group?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to self-intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.
...

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**1**answer

116 views

### Semidirect products with braid groups and type $F_\infty$

Let $F$ be a group which is strongly type $F_\infty$ in the sense that every subgroup is of type $F_\infty$. Here, type $F_\infty$ means that the group admits a classifying space with compact skeleta.
...

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164 views

### Computable link invariants

I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like here). For a braid $B$, we know how to represent it in ...

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206 views

### What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$....

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**1**answer

124 views

### An analogue of cabling for configuration spaces

There is a well-known operation known as cabling for knots, and also for braid groups, where it is a homomorphism
$$\beta_k \times \beta_\ell \longrightarrow \beta_{k\ell}$$
given by thickening up the ...

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votes

**0**answers

97 views

### Pure braid groups of the complement of a lattice in the complex plane: generators and relations

Where can I find a presentation (by `natural' generators and relations between them)
of the pure braid groups $PB_n(S)$ (for $n>0$) of $S=\mathbb C\setminus (\mathbb Z\oplus i \mathbb Z)$?
Thanks ...

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**1**answer

229 views

### The homology of the braid group with coefficients in the Burau representation

Let $B_n$ denote the braid group with $n$ braids. The Burau representation $B_n\to GL_n(\mathbb{Z}[t^{\pm1}])$ makes $(\mathbb{Q}[t^{\pm1}])^n$ a $B_n$-module. I am curious in knowing what $H_i(B_n, (\...