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4
votes
0answers
89 views

Braid relations $n_\alpha n_\beta n_\alpha \ldots = n_\beta n_\alpha n_\beta \ldots $ in arbitrary reductive groups

I'm currently trying to prove or disprove the following claim. First let me set up some notation. Let $G$ be a connected reductive group over a field $K$, let $S \leq Z \leq N \leq G$ be respectively ...
5
votes
1answer
142 views

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 ?
-1
votes
1answer
94 views

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 ...
2
votes
0answers
73 views

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 ...
2
votes
0answers
103 views

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 ...
2
votes
1answer
86 views

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} ...
2
votes
2answers
160 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 ...
0
votes
0answers
54 views

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. ...
3
votes
1answer
145 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 ...
0
votes
0answers
79 views

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 $M$. ...
5
votes
0answers
276 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.) ...
6
votes
1answer
283 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 ...
3
votes
1answer
125 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 ...
6
votes
1answer
221 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 $A_{j,k}$ ...
0
votes
0answers
20 views

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 ...
0
votes
1answer
141 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 ...
2
votes
1answer
114 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 ...
1
vote
0answers
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 ...
2
votes
1answer
146 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 ...
4
votes
1answer
202 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 ...
4
votes
1answer
136 views

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 ...
0
votes
1answer
121 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 $n-1$ ...
0
votes
0answers
82 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 ...
2
votes
0answers
141 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 ...
6
votes
0answers
281 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. ...
1
vote
1answer
107 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. ...
2
votes
1answer
114 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 ...
13
votes
0answers
190 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 ...
6
votes
1answer
118 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 ...
2
votes
0answers
94 views

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 ...
8
votes
0answers
150 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, ...
4
votes
0answers
161 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 ...
10
votes
0answers
209 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 ...
8
votes
0answers
160 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 $I^3$, ...
4
votes
1answer
286 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 ...
10
votes
2answers
316 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 $H^*(Br_n, \lambda)$?
3
votes
1answer
221 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 ...
2
votes
1answer
144 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 ...
13
votes
0answers
286 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 ...
8
votes
1answer
403 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 ...
13
votes
3answers
494 views

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 ...
2
votes
1answer
187 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 ...
7
votes
0answers
162 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 ...
15
votes
4answers
854 views

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 ...
6
votes
2answers
208 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 ...
5
votes
1answer
212 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 ...
20
votes
2answers
648 views

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 ...
1
vote
0answers
303 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 ...
4
votes
1answer
341 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 ...
7
votes
1answer
401 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 ...