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4
votes
0answers
92 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 ...
8
votes
0answers
129 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 ...
6
votes
0answers
108 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
232 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
295 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
153 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
114 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 ...
12
votes
0answers
248 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
265 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
419 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
152 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
145 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 ...
14
votes
4answers
788 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 ...
5
votes
2answers
193 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
170 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
563 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
219 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
0answers
226 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 ...
6
votes
1answer
313 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 ...
3
votes
0answers
302 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 ...
4
votes
0answers
184 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 ...
11
votes
1answer
353 views

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 $$B_n \to GL_{n \choose 2} \mathbb Z[t^{\pm}, q^{\pm}] $$ ...
3
votes
0answers
158 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 ...
4
votes
2answers
796 views

The word problem in braid groups

I have read a statement from Sossinsky and Prasolov' s book "Knots, L─░nks, Braids and 3-Manifold", it says that two reduced word represent isotopic braids if and only if they have the same reduced ...
2
votes
1answer
233 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 ...
8
votes
1answer
532 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 ...
4
votes
2answers
577 views

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 ...
4
votes
1answer
342 views

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 ...
5
votes
1answer
464 views

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 ...
3
votes
1answer
505 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 ...
0
votes
1answer
252 views

Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid?

It is a known result that if $B$ is an $n$ braid over a disk, then $B$ naturally induces an isomorphism between the fundamental group of a disk with n points removed and the fundamental group of the ...
3
votes
0answers
334 views

Positivity of braid monodromy of curve singularities

I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to ...
15
votes
0answers
516 views

Splitting of homomorphism from cactus group to permutation group

We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...
4
votes
0answers
173 views

restricting “dances of minimal cost” (optimization on braids/permutations?)

Consider applying the permutation (1,3,0,5,2,7,4,6) to the integers (0,1,2,3,4,5,6,7) three times. I call this a "dance of minimal cost" because all unordered pairs in {0..7} meet each other, and the ...
3
votes
3answers
804 views

Normal subgroups of braid groups

There is a lot of normal subgroups in braid groups (for example there is an action of $B_n$ on unitriangular bilinear forms on $R^n$ over arbitrary commutative ring $R$: $b_i\colon e_j\mapsto e_j$, ...
0
votes
0answers
283 views

Properties of the Jones polynomial

Let $V(L)$ be the Jones polynomial of the oriented link $L$. For $\alpha \in B_n$, we write $V(\alpha)$ for $V(\hat{\alpha})$, where $\hat{\alpha}$ is the closed braid associated to $\alpha$. The ...
11
votes
1answer
536 views

Realizing braid group by homeomorphisms

Markovich and Saric proved the following remarkable theorem. Let $S$ be a compact surface of genus at least $2$ and let $MCG(S)=\pi_0(Homeo^{+}(S))$ be the mapping class group of $S$. There is then ...
1
vote
1answer
540 views

Markov Trace and Markov Property

Hey guys, I'm a computer science student attempting to understand a quantum algorithm that uses braid theory - something I'm completely unfamiliar. I've getting through the algorithm but I can't ...
1
vote
1answer
216 views

Simplified Jones trace invariant for links

Jones (1985) defines a simplified trace invariant for knots by $W_K(t)=\frac{1-V_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W_K(i)$. Does this work for oriented links as ...
0
votes
1answer
220 views

Are braid links proper links?

Are braid links proper links? Or are the concepts involved unrelated?
4
votes
1answer
290 views

Algebraic K-groups and braids

This is (I think) a reference request: Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
8
votes
1answer
562 views

Cohomology of a configuration space

The symmetric group $\Sigma_k$ acts on $X=F({\mathbb R}^n,k)$, the ordered configuration space of $k$ points in ${\mathbb R}^n$. If $n$ is odd, the cohomology $H^*(X;{\mathbb Q})$ is a rank-one ...
7
votes
1answer
298 views

Aspherical homotopy orbit space of configurations on the 2-sphere

The group SO(3) acts naturally on $S^2$ and thus on $Conf(S^2, q)$, the configuration space of $q$ distinct points on the 2-sphere, via the diagonal action on $S^2 \times...\times S^2$. This is a ...
8
votes
2answers
517 views

Borromean braids

Consider the Kernel $K_n$ of the natural group homomorphism from the $n$-th braid group to the symmetric group. Then one can delete the $m$-th braid. This is a well defined homomorphism ...
18
votes
6answers
1k views

Is there a q-analog to the braid group?

The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups: $$ 1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$ where $S_n$ is the symmetric ...
14
votes
1answer
518 views

Loop spaces and infinite braids

The Artin braid groups $B_n$ and the symmetric groups $S_n$ are closely related by the maps $1 \to P_n \to B_n \to S_n \to 1$. The infinite symmetric group has interesting interactions with homotopy ...
3
votes
1answer
223 views

Is a positive link the closure of a positive braid?

Alexander's Theorem guarantees that every oriented link is the closure of some braid. In other words, the map $$ \displaystyle \coprod\_n \mathcal B_n\longrightarrow \{\text{ oriented links }\} $$ ...
11
votes
4answers
436 views

Minimal-length embeddings of braids into R^3 with fixed endpoints

(Apologies in advance for any imprecision in the following; I am a computer scientist and regret never having taken an actual course on topology.) One way to define the pure braid group $P_n$ is as ...
2
votes
2answers
391 views

Presentation for an infinite index subgroup of the braid group

If $H$ is an infinite index subgroup of the braid group $\mathcal{B}_n$, is there a way to find a presentation for $H$ ?
12
votes
5answers
1k views

Is the pure braid group on three strands generated as a normal subgroup of the braid group by the six-crossing braid?

Artin's presentation of braid group on three strands is: $$ B_3 = \langle l,r : lrl = rlr \rangle $$ where you should think of "$l$" as the positive crossing between the left and middle strands and ...