The braid-groups tag has no wiki summary.

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### 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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107 views

### 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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204 views

### 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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**1**answer

92 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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**1**answer

104 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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262 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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134 views

### (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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176 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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105 views

### 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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76 views

### 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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120 views

### 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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259 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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99 views

### 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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96 views

### 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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174 views

### 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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111 views

### 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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121 views

### 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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148 views

### 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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179 views

### 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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150 views

### 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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265 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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308 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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189 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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132 views

### 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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265 views

### 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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**1**answer

333 views

### 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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170 views

### 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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154 views

### 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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206 views

### 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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189 views

### 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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274 views

### 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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278 views

### 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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364 views

### 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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321 views

### 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 ...

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207 views

### Braid group action on canonical basis

This is a question about Lusztig's theory of based modules. This theory is elementary but far from easy and is developed in Chapter 27 of "Introduction to quantum groups".
Let $V$ be a highest weight ...

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### Lower bounds on dimensions of faithful representations of braid groups

Let $B_n$ be the braid group on $n$ strands. It's a theorem of Daan Krammer and Stephen Bigelow that there is a a faithful representation
$$B_n \to GL_{n \choose 2} \mathbb Z[t^{\pm}, q^{\pm}] $$
...

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172 views

### Constructing Markov traces simply

Short version: I wondering how to simply check if a proposed Markov trace, $\phi$ had the correct property using techniques similar to those from the Akutsu-Wadati 1987 paper `Exactly solvable models ...

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### The word problem in braid groups

I have read a statement from Sossinsky and Prasolov' s book "Knots, Lİnks, Braids and 3-Manifold", it says that two reduced word represent isotopic braids if and only if they have the same reduced ...

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247 views

### Computing an Invariant for Knots via Braid Words?

I've been reading up on Knot Theory (which is not my area of expertise) and am stuck in the following bit of logic:
Statement 1: Every knot can be represented as a braid.
Statement 2: There's a ...

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578 views

### Orbifold fundamental group and configuration space

Hi,
I'm not very familiar with (even simple examples of) orbifolds, so my first question is:
Let $C_2$ be $\mathbb{C}$ with one cone singularity at 0 of index 2. What is the fundamental group of ...

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### Presentation of the pure Artin groups

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by
$$W= < T_1, \dots, T_n | T_i^2=1, T_iT_jT_i \ldots = T_jT_iT_j \ldots, i \neq ...

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### How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?

The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...

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### Jones Polynomial of the trace closure of the fundamental braid

The fundamental braid $\Delta_n \in B_n$ is simply a twist by $\pi$ applied to the entire row of $n$ strands. In terms of Artin generators, it is given by
$$
\Delta_n = (\sigma_1 \sigma_2 \cdots ...

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549 views

### Almost-direct product and 1-formality

Hi everyone,
Let $G$ be a finitely presented group. To $G$ is associated in a functorial way a Malcev Lie algebra which can be constructed in several equivalent ways. Roughly speaking, it is the ...