# Questions tagged [braid-groups]

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158
questions

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votes

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191 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

**1**answer

264 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 $n$ distinct ...

**2**

votes

**0**answers

76 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

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231 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 $A_{12} \in \Sigma_b$ ...

**7**

votes

**5**answers

688 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: ...

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votes

**0**answers

98 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\...

**1**

vote

**1**answer

141 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 $...

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votes

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72 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, ...

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votes

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457 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 ...

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99 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_{\...

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votes

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96 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 $i>0$.
...

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votes

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87 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 \...

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vote

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181 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 ...

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127 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|...

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votes

**1**answer

161 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 ...

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votes

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79 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 ...

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votes

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86 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})\...

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votes

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126 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_{...

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votes

**2**answers

256 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 $s^2 = t$ ...

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98 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^{\...

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votes

**1**answer

146 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

**1**answer

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 ...

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votes

**1**answer

426 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

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267 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 ...

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votes

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140 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

**1**answer

232 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_{...

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votes

**1**answer

290 views

### Centralizer of a generator in a braid group

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

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votes

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100 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

**1**answer

254 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

**1**answer

561 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

**1**answer

455 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

**1**answer

386 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 ...

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votes

**0**answers

97 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 ...

**5**

votes

**2**answers

216 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) \...

**1**

vote

**1**answer

101 views

### Orderable subgroup of the braid groups over the 2-sphere

$$B_{n}(S^2)=\langle \sigma_1,\sigma_2,...\sigma_{n-1}\mid
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i} \text{ if } |i-j|>1;\qquad$$ $$\qquad
\sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{...

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votes

**0**answers

51 views

### Question about the mutation of a cluster seed associated to any word of the braid semigroup

Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ ...

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votes

**2**answers

361 views

### Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?

If $X$ is a topological space, write $C_n(X)$ for the configuration space of distinct ordered tuples of points in $X$:
$$C_n(X) = \{(x_1, \ldots, x_n) \in X^n \mbox{ so that $i \neq j \implies x_i \...

**10**

votes

**0**answers

205 views

### What is the preimage of a braid in a covering space branched over the braid?

For a knot $K\subset \mathbb{S}^3$, one can construct the covering space branched over that knot by assigning elements of the symmetric group $S_n$ to each arc of the knot. You can find the knot group ...

**3**

votes

**1**answer

133 views

### The singularity type of a non-torus link

It is a well-know result that singular surfaces like
$$x^p+y^q=0$$
(for complex $x$ and $y$) can be associated with $(p,q)$-torus links by considering the intersection of this surface with a small ...

**6**

votes

**1**answer

181 views

### How to braid a ribbon knot

Is there any algorithm known for braiding ribbon knots? More specifically I need to braid a generic ribbon knot presented as boundary of a ribbon surface= union of two 0-handles and one 1-handle. (...

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votes

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232 views

### Is conjugacy problem hard in braid group?

Recently I studied the braid group and conjugacy problem. It is believed that conjugacy problem is hard on braid group. My friend gave me an EXE file, and I use it for solving conjugacy problem, as an ...

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vote

**0**answers

139 views

### How to show the determinant of $B - I$ is zero? [closed]

Let $n \geq 2$ be a positive integer and
$$\beta_i= \left(
\begin{array}{c|c c|c}
I_{i-1} & 0 & 0 & 0\\
\hline
0 & 1-q & q & 0\\
0 & 1 & 0 & 0\\
\hline
...

**2**

votes

**1**answer

162 views

### Plat representations of unlinks

Suppose that $\beta$ is a $2n$-strand braid with plat closure $L$. We can multiply $\beta$ on either side by a member of the Hilden subgroup to get a new braid whose plat closure is still $L$. Or we ...

**4**

votes

**1**answer

238 views

### fundamental group of configuration spaces of ordered points on open Riemann surfaces

Let $\bar{X}$ be a compact Riemann surface of genus $g>0$. Let $X$ be $\bar{X}$ minus a finite set of points $\{a_1,\ldots,a_n\}$ ($n\geq 1$). Let $X^{(r)}$ be the configuration space of $r$ ...

**3**

votes

**0**answers

120 views

### Does the notion of a critical point extend from set theory to Braid groups?

Let $B_{\infty}$ denote the infinite strand braid group. Let $\text{sh}:B_{\infty}\rightarrow B_{\infty}$ be the homomorphism defined by $\text{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Give $...

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vote

**0**answers

105 views

### Is there an analog of Reidemeister's theorem for braids in a surface?

Reidemeister's classical theorem describes the set of links in $\mathbb R^3$ up to isotopy as the set $\{ \textrm{diagrams in } \mathbb R^2\textrm{ with crossings}\}$ modulo certain local relations on ...

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votes

**0**answers

55 views

### Is the word problem in the braid group quotient $B_{n}/N$ solvable where $N$ is the normal subgroup generated by conjugates of $\sigma_{i}^{2r}$?

Let $r\geq 2$. Let $N$ be the normal subgroup of $B_{n}$ generated by conjugates of $\sigma_{i}^{2r}$. Then is the word problem in the quotient group $B_{n}/N$ solvable (in polynomial time)? ...

**1**

vote

**1**answer

221 views

### How quickly can one compute the Hurwitz action of braid groups on finite groups?

Let $G$ be a finite group. Define the Hurwitz action of $B_{n}$ on $G^{n}$ by letting
$(x_{1},...,x_{n})\sigma_{i}=(x_{1},...,x_{i}x_{i+1}x_{i}^{-1},x_{i},x_{i+2},...,x_{n})$. I wonder what algorithms ...

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336 views

### Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$

This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...

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votes

**0**answers

106 views

### Krammer representation of an essential braid

A braid element $b$ in the braid Group $B_n$ is an essential braid if it does not have at least one of the generators in it. For example, $b=\sigma_1\sigma_2\sigma_4 \in B_5$ is essential since $\...