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### 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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### unordered configuration space of pointed space

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

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### 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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### 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 ...

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### 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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154 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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### 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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142 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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### cohomology of labelled configuration space & relation with braid space

Let $M$ be a manifold and $(X,*)$ be a pointed topological space. ( If we want, we can let $M=S^2,S^1\times \mathbb{R},etc.$)
Let $F(M,k)$ be the ordered configuration space of $k$-tuples on $M$.
...

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270 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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278 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 ...

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### 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 ...

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### 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 $A_{j,k}$ ...

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### Action as specified in the extension given by the holomorph

I am reading Braid Groups, Free Groups, and The Loop Space of the 2-sphere
by F.R. Cohen and J. Wu, which is available here.
Here is an extract of the paper:
(Please refer to the paper for ...

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127 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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113 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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268 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
...

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### (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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197 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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### 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 ...

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### 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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### Normal Form of Homotopy Pure Braids?

It is well known that a pure braid has a normal form (also called the combed form). Namely, let $P_n$ be the set of pure braids of $n$ strands and let $d_i:P_n\to P_{n-1}$ be the $i$th "forgetting ...

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### 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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278 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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, ...

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### The Alexander-Conway polynomial: from knots to braids?

The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...

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### The Markov trace via Bott-Samelson fibers?

Let $H_n$ be the Hecke algebra of GL(n), i.e., the algebra over $\mathbb{Q}(q)$ with generators $T_1,
\ldots, T_{n-1}$ which satisfy the braid relations and also $T^2 = (q-1) T + q$.
Recall the ...

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### Does the shortest path between two braids pass through string links?

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.
This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...

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282 views

### word problem for the fundamental group of complements

It is well known that the finite type (pure) Artin groups have solvable word problem. This was proved by Deligne in 1972. His aim was to show that the complement of a simplicial hyperplane arrangement ...

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314 views

### H*(braid group, irrep of symmetric group) = ?

As in the title, say $\lambda$ is some irrep of the symmetric group $S_n$, and $Br_n$ the braid group on $n$ strands,
What is $H^*(Br_n, \lambda)$?

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211 views

### Finite generation of the commutator subgroup of the pure braid group

Let $PB_n$ be the pure braid group on $n$ strands. The group $PB_n$ has every conceivable finiteness property. Also, it has a large abelianization. My question is whether the commutator subgroup ...

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### Braided coverings and braided monodromy

We can map from set of coverings over $X$ to symmetric group $\mathfrak{S}_n$ via monodromy (if we fix a loop at the basepoint). Also we can consider braid group $Br_n(Y)$, allow strands pass through ...

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### Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE.
There exists a rather remarkable ...

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### When do two positive braids represent the same link?

Let $B_n$ be the braid group on $n$ strands, with the usual generators: $s_1, \ldots, s_{n-1}$ and their inverses, where $s_i$ is a positive half-twist interchanging the strands labelled $i$ and ...

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### Does this subgroup of “even braids” have a name?

The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation ...

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### holomorphic automorphisms of universal cover of configuration spaces

Hello everyone,
I have been trying (without success) to determine the following. Let $P$ denote the space of monic polynomials of degree $n$ with complex coefficients, which have distinct roots. It ...

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### Does the braid group act faithfully on the quantized enveloping algebra?

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where ...

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### Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question was originally posted on math.SE by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was made in ...

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### Automatic factors of braid groups

Consider the braid group $B_n$ presented in terms of the usual Artin generators $\sigma_1,\ldots,\sigma_{n-1}$. Now add the additional relations $\sigma_i^k = 1$ for $i=1,\ldots,n-1$. For lack of a ...

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### Relations in a particular subgroup of the braid group.

I think this should be a 10 minute exercise in a decent computer algebra package - unfortunately I'm hopelessly ignorant of such things, so I'm putting it up here in the hope that someone will be kind ...

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### The image of the point-pushing group in the hyperelliptic representation of the braid group

Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation
$\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$
called the "hyperelliptic representation," which ...

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### Explicit Computations of Dynkin Diagrams of Isolated Singularities

Let $f$ be a complex polynomial with an isolated singularity at the origin. Take a Morse deformation $\tilde{f}$, and consider the braid group action on the set of distinguished bases of vanishing ...

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### Decomposition of Braid Groups

I've been trying to google this question, but to no avail. The question sounds elementary but I hope it's suitable for the experts at MO!
Let $B_n$ be the braid group on $n$-strands and $P_n$ be the ...

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### Resemblance between Birman-Murakami-Wenzl algebra representations and the Lawrence-Krammer representations

At the end of Stephen Bigelow's paper "Braid Groups are Linear", he mentioned that there is a striking resemblance between the matrices of the Lawrence-Krammer (LK) representations and "those of a ...

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### Galois groups and braid groups [closed]

Braid group can be viewed as a symmetry group with a "one more dimension to pass through". Is there any "Galois theory", where the braid groups plays analoguos role as a symmetry groups in a native ...