Questions tagged [braid-groups]
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213
questions
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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
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72
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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
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330
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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
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1
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158
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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
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1
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255
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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
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292
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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 $n$ ...
4
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2
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284
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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 ...
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343
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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 $A_{12} \in \Sigma_b$ ...
9
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5
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1k
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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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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
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2
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354
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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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0
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107
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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
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587
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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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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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102
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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
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96
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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
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217
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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
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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
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243
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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 ...
6
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105
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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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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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140
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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_{...
7
votes
2
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450
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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
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113
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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$$ which gives us an action of the braid group $B_n$ on $V^{\...
3
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203
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Basis for Annular Skein Algebra
Background/Notation:
Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis
$\{T_{w}\}_{w\in S_{...
6
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149
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Ordinary differential operators satisfying braid relation?
Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...
12
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491
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Formality of the 2nd ordered configuration space of a closed Riemann surface
If $X$ is a smooth manifold, we define its kth ordered configuration space as $$F_kX:=\{(x_1, \ldots,x_k) \; | \; x_i \neq x_j \,\, \mathrm{if} \, \, i \neq j\},$$
in other words, $F_kX = X^k - \Delta,...
6
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311
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Epimorphisms from the genus $2$ surface braid group to finite groups
This question is somehow related to my previous MO question Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$; for the reader convenience, let me write down again the relevant ...
9
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184
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Interactions between pseudoline arrangements and braid groups?
It is common to represent
pseudoline arrangements
as wiring diagrams:
Fig. from: "Hamiltonicity and colorings of arrangement ...
7
votes
1
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265
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Homotopy type of the semi-simplicial set of symmetric groups
Consider the collection of symmetric groups $\{\Sigma_n\}_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma_{...
3
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1
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415
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Centralizer of a generator in a braid group
Given a braid group
$$
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
$$...
5
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168
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Question about terminology, and reference request related to the braid operad
Let $\Delta_n$ stand for the Garside element of the braid group $B_n$. It turns out that the family of all Garside elements have the following ``operadic'' property:
$$
\Delta_n\left[ \Delta_{k_1},\...
8
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1
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274
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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
$...
22
votes
1
answer
699
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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
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1
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608
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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
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697
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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
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126
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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
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2
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249
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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
$$M^n \backslash \{ (x_1, \ldots, x_n) \...
1
vote
1
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129
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Orderable subgroup of the braid groups over the 2-sphere
$$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
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0
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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
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2
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412
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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 $X$:
$$C_n(X) = \{(x_1, \ldots, x_n) \in X^n \mbox{ so that $i \neq j \implies x_i \...
10
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0
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241
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What is the preimage of a braid in a covering space branched over the braid?
For a knot $K\subset \mathbb{S}^3$, one can construct the covering space branched over that knot by assigning elements of the symmetric group $S_n$ to each arc of the knot. You can find the knot group ...
3
votes
1
answer
262
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The singularity type of a non-torus link
It is a well-know result that singular surfaces like
$$x^p+y^q=0$$
(for complex $x$ and $y$) can be associated with $(p,q)$-torus links by considering the intersection of this surface with a small ...
6
votes
1
answer
250
views
How to braid a ribbon knot
Is there any algorithm known for braiding ribbon knots? More specifically I need to braid a generic ribbon knot presented as boundary of a ribbon surface= union of two 0-handles and one 1-handle. (...
2
votes
0
answers
256
views
Is conjugacy problem hard in braid group?
Recently I studied the braid group and conjugacy problem. It is believed that conjugacy problem is hard on braid group. My friend gave me an EXE file, and I use it for solving conjugacy problem, as an ...
1
vote
0
answers
143
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How to show the determinant of $B - I$ is zero? [closed]
Let $n \geq 2$ be a positive integer and
$$\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
266
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
301
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
128
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 $...
2
votes
0
answers
126
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 ...