Questions tagged [braid-groups]

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5
votes
1answer
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
1answer
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
0answers
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
0answers
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
5answers
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: ...
6
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0answers
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
1answer
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 $...
2
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0answers
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, ...
2
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0answers
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 ...
2
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0answers
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_{\...
2
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0answers
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 $i>0$. ...
3
votes
0answers
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 \...
1
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0answers
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 ...
4
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0answers
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|...
5
votes
1answer
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 ...
5
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0answers
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 ...
3
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0answers
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})\...
4
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0answers
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_{...
4
votes
2answers
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 $s^2 = t$ ...
4
votes
0answers
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^{\...
2
votes
1answer
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
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
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
2answers
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 ...
9
votes
0answers
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
1answer
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_{...
3
votes
1answer
290 views

Centralizer of a generator in a braid group

Given a braid group $$ B_n \simeq \left\langle x_1,\ldots,x_{n-1} \middle| \begin{array}{l} x_ix_j = x_jx_i, \;\text{for } |i-j|>1 \\ x_ix_{i+1}x_i = x_{i+1}x_ix_{i+1} \end{array} \right\rangle $$...
5
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0answers
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
1answer
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
1answer
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
1answer
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
1answer
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 ...
5
votes
0answers
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
2answers
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
1answer
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_{...
2
votes
0answers
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 $\prod$ ...
10
votes
2answers
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
0answers
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
1answer
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
1answer
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. (...
2
votes
0answers
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 ...
1
vote
0answers
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
1answer
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
1answer
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
0answers
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 $...
1
vote
0answers
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 ...
2
votes
0answers
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
1answer
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 ...
9
votes
0answers
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 ...
2
votes
0answers
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 $\...