# Questions tagged [braid-groups]

The braid-groups tag has no usage guidance.

189
questions

8
votes

2
answers

227
views

### Coherence theorem in braided monoidal categories

In MacLane's Categories for the working mathematician, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}_{\mathrm{BMC}}(B,M)\simeq M_0$ where $B$ is the braid ...

19
votes

0
answers

303
views

### Are braid groups known to not be linear over $\mathbb{Z}$?

$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This ...

5
votes

0
answers

150
views

### Coxeter's braid group quotients

Coxeter's result is that if the generators of the braid group $B_n$ on $n$ strands fulfill a relation $\forall_i\sigma_i^k=1$, then $1/n+1/k>1/2$ must hold to get a finite quotient of $B_n$. In ...

3
votes

1
answer

298
views

### Dehornoy's proof that the application of two elementary embeddings is an elementary embedding

What is meant by the statement and the proof of Lemma 3.2 in Chapter XII of Dehornoy's book Braids and Self-Distributivity?
That lemma states "Assume that $j_1$ and $j_2$ are elementary ...

2
votes

0
answers

76
views

### Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\cd{cd}$Let $B_{k}(S_{g}),$ $\MCG(S_{g};k)$ and $\MCG(S_{g}))$ denote the braid group, the mapping class group (relative ...

5
votes

1
answer

266
views

### Representation stability for systems of braid group representations

In general, if I understand correctly, the representation theory of the braid groups is quite complicated, and there's no classification of the irreducibles. However, the braid groups form a sort of ...

3
votes

1
answer

199
views

### Good algorithmic properties for quotients of braid groups

I'm trying to understand some things about quotients of braid groups, and particularly I'd like to solve the word problem for some elements of these quotients. I'm using MAGMA to try to access this, ...

8
votes

0
answers

119
views

### Configuration space of 4 points as an orbifold

Setup: Consider the braid group $B_n$. One way to define this is as the fundamental group of the unordered configuration space $UC_n(\mathbb{C}) = \{\{z_1,\dotsc,z_n\}\subset \mathbb{C} \mid z_i \not= ...

3
votes

1
answer

51
views

### Arc complex characterization of braids with trivial closure

A braid $\beta\in B_n,$ the braid group with $n$ strands, viewed as the mapping class group $\mathrm{Mod}(\mathbb{D}_n)$ of the disk with $n$ punctures is trivial if and only if $\beta$ acts trivially ...

2
votes

0
answers

195
views

### Braid group and the fundamental group of coefficients of polynomials

I am studying Norbert A'Campo's article "Tresses, monodromie et le groupe symplectique" and I am trying to understand why $\rho(t_i)=T_i$. I am stating this just for the sake of reference, ...

2
votes

0
answers

36
views

### Finiteness of almost maximal submonoids in positive braids

A numerical semigroup of genus $g$ is a submonoid $S=S+S$ of $\mathbb N$ missing $g$ elements (gaps) of $\mathbb N$. It is not hard to show that
there are only finitely many numerical semigroups of ...

2
votes

0
answers

75
views

### Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?

As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms.
There are a relationship between the mapping class ...

2
votes

0
answers

120
views

### Braid 2-groups, symmetric 2-groups

Is there an object which can be called a "braid 2-group"? Or a "symmetric 2-group"? (Note: not a braided 2-group)
I am ignorant about 2-categories but I hope that a good candidate ...

5
votes

1
answer

115
views

### Composite knots and their braid words

Given a composite knot K = K_1 # K_2, I wonder how the braid word looks like. Is it possible to see from the word that the knot is composite?
I am not aware of a statement such as "the closure of ...

8
votes

1
answer

201
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

1
answer

162
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

1
answer

78
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

0
answers

157
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 ...

8
votes

2
answers

466
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 ...

12
votes

1
answer

161
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

0
answers

46
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

1
answer

74
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

2
answers

699
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

1
answer

212
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

2
answers

282
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

1
answer

192
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

0
answers

86
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

0
answers

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

0
answers

244
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

1
answer

119
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

1
answer

234
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

283
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

0
answers

103
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

0
answers

288
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

5
answers

886
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

0
answers

116
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

2
answers

282
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

0
answers

96
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

0
answers

579
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

0
answers

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

0
answers

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

0
answers

89
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

0
answers

205
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

0
answers

137
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

1
answer

205
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

0
answers

91
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

0
answers

95
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

0
answers

132
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

2
answers

345
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

0
answers

110
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^{\...