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Questions tagged [braid-groups]

Should be used with the (group-theory) tag. For questions about braid groups: groups which arise as fundamental groups of configuration spaces and formalize the study of the everyday notion of a braid.

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$P_{n}$ is the minimal normal subgroup of $B_{n}$ containing $\sigma_{1}^{2}$ is $ A_{1,2}$

I was reading Braid Groups by Kassel and Turaev, I need to show the following: $P_{n}$ is the minimal normal subgroup of $B_{n}$ containing $\sigma_{1}^{2}$ is $ A_{1,2}$. I know there exists a ...
Dwaipayan Sharma's user avatar
3 votes
2 answers
87 views

Subgroup of braid group $B_3$ isomorphic to itself

Consider the braid group $$B_3=\langle x,y:xyx=yxy\rangle.$$ It has a proper subgroup $N$, defined as follows: $g$ is in $N$ if and only if the sum of all exponents in $$g=\prod u_i^{\varepsilon_i},\ ...
atzlt's user avatar
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0 answers
9 views

Need help visualizing an n-braid formed by connecting punctured planes with strings

I was reading Princeton companion and I got the following at page no.160 Take two parallel planes, each punctured at n points. "Label the holes 1 to n in each plane, and run a string from each ...
Abhinav Patel's user avatar
2 votes
0 answers
42 views

Reference Request: Visual Approach to Symmetric Groups

The symmetric group is a factor of the braid group (see e.g. Surjective Group Homomorphism From Braid Group Into Symmetric Group, Symmetric group, Braid Groups, and related groups). Consequently one ...
Alp Uzman's user avatar
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1 vote
1 answer
36 views

Injection from bivariate Laurent polynomials into the complex numbers

I've been reading about the Lawrence-Krammer representation of the braid group, which is a linear representation over the ring of bivariate Laurent polynomials $\mathbb{Z}[t, t^{-1}, q, q^{-1}]$. I've ...
Tuomas Laakkonen's user avatar
0 votes
1 answer
85 views

Relationship between braid group $ B_n $ and mapping class group of punctured spheres

The braid group on $ n $ strands, $ B_n $, is the mapping class group of the disk with $ n $ punctures. Let $ \Sigma_{g,n} $ denote an oriented surface of genus $ g $ with $ n $ punctures. Let $ M_{g,...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
53 views

Relationship between braid group and $ MCG(\Sigma_g) $

Let $ B_n $ be the braid group on $ n $ strands. Let $ MCG(\Sigma_g) $ be the mapping class group of a surface of genus $ g $. Is there a relationship between $ B_{2g+1} $ and $ MCG(\Sigma_g) $? For $ ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
88 views

In what sense is the Temperley-Lieb algebra related to the Braid group?

Question: In what sense is the Temperley-Lieb algebra $ TL_n $ related to (a representation of?) the braid group $ B_n $? For example, is $ TL_n $ the algebra of matrices generated by the image of $ ...
Ian Gershon Teixeira's user avatar
3 votes
0 answers
59 views

Action of braid groups on regular trees

Question: Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive ...
Ian Gershon Teixeira's user avatar
8 votes
1 answer
224 views

Is every finite simple group a quotient of a braid group?

Question: Is every finite simple group a quotient of a braid group? Context: The braid group on two strands $ B_2 $ is isomorphic to $ \mathbb{Z} $ and so the infinite family of abelian finite simple ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
54 views

Does a Braid group/ more generally a finitely generated group have finitely many perfect quotients?

I was wondering about perfect quotients of the Braid groups $ B_n $. Certainly $ B_1=1 $ and $ B_2=\mathbb{Z} $ have finitely many perfect quotients. For $ B_3 $, $$ 1 \to Z(B_3)=\mathbb{Z} \to B_3 \...
Ian Gershon Teixeira's user avatar
3 votes
0 answers
61 views

Equivalence of two presentations for a quotient of the braid group

I'm considering two presentations for quotients of the 3-strand braid group $B_3$ that I believe should be equivalent (i.e. yield the same quotient). There is an integer parameter $d \ge 7$ involved, ...
Ethan Dlugie's user avatar
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4 votes
1 answer
188 views

What is the order of the braid group with n strands, on the 2-sphere?

Consider the the braid group of $n$ strands on the 2-sphere. Visually, this is a braid between two concentric circles. How many different braids are there for a given $n$? I have tried drawing the ...
SAB's user avatar
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0 answers
62 views

In the coherence theorem for braided monoidal categories, why does it suffice to show the result for strict monoidal categories only?

$\newcommand{\M}{\mathcal{M}}\newcommand{\B}{\mathfrak{B}}\newcommand{\hom}{\operatorname{Hom}}\newcommand{\BM}{\mathsf{BM}}\newcommand{\SBM}{\mathsf{SBM}}\newcommand{\S}{\mathcal{S}}$I refer to the ...
FShrike's user avatar
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Generators of the monodromy generating braid groups

I'm reading the article 'Complex reflection groups, braid groups and Hecke algebras', by Broué, Malle and Rouquier, but I need some help with the 'generators of the monodromy' they defined and that ...
cgu's user avatar
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6 votes
2 answers
291 views

Braid group center intuition

Braid groups have an infinite cyclic group center, generated by the square of the fundamental braid. Geometrically, the fundamental braid has the property that any two strands cross positively exactly ...
Single Malt's user avatar
0 votes
0 answers
43 views

Braid collapsed to 2 distinguished points

Is there a study of non crossing paths from point A to point B in 3-space? Could be braid theory but in that theory the strands don't all converge to the same point(s). The idea is equivalent to ...
zeta space's user avatar
1 vote
1 answer
110 views

Braid groups for ropes?

What is the proper braid theoretic treatment of rope construction? Any are there any good references/sources? It would seem that they are something like the Artin braid group, but instead of operating ...
JustAskin's user avatar
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1 answer
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Symplectic representation of the braid group

Given a natural number $g$, there is a natural representation $R_{g}:B_{2g+1}\to\text{Sp}(2g,\mathbb{Z})$ of the braid group on $2g+1$ threads within the $2g\times 2g$ symplectic matrices, known as ...
Bob Knighton's user avatar
0 votes
0 answers
46 views

How do I prove this expression for the q-deformed factorial?

Kauffman states without proof in "Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds" that these 2 expressions for the q-deformed factorial are equal: $$ [n]! = \prod_{k=1}^n \...
Joel Mills's user avatar
2 votes
1 answer
298 views

Half-twist (Dehn twist) on braids represented as elements of the mapping class group on an n-punctures disk

Let $\mathbb{D}_n$ denote a disk with $n$ marked points or punctures. The mapping class group of $\mathbb{D}_n$ is isomorphic to the braid group $B_n$. Elements of the mapping class group of $\mathbb{...
user523692's user avatar
1 vote
2 answers
146 views

how did they come up with polynomials as knot invariant

I understand how to calculate a Jones polynomial for a given knot, but I am not sure why would one search for a polynomial for invariants. How did he come up with these calculation rules?
wooohooo's user avatar
  • 176
1 vote
0 answers
92 views

Braided coherence 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 $\text{hom}_{BMC}(B,M)\simeq M_0$ where $B$ is the braid category, $M$...
QGM's user avatar
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1 vote
0 answers
49 views

Why is Conway notation relevant, instead of braid notation (for knots)

In the early XXth century, Alexander showed that every knot is the closure of a braid, and Markov gave necessary and sufficient conditions for two braids to have the same closure. My question is: why ...
Gutiérrez's user avatar
1 vote
0 answers
69 views

$s$-isotopy of braids is an equivalence relaiton

So, I have been reading Chapter 6 of "A Study of Braids" by Murasugi and Kurpita. The main goal of the chapter is to prove that many notions of equivalence of braids actually coincide. I ...
ConcreteSwanDive's user avatar
4 votes
0 answers
55 views

Index of pure braid group

The index of a subgroup $H$ in a group $G$ is the number of left cosets of $H$ in G, or equivalently, the number of right cosets of $H$ in $G$. The index is denoted $(G:H).$ Let $S$ be a surface. If $...
King Khan's user avatar
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4 votes
0 answers
57 views

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

Let $B_{k}(S_{g}),$ $MCG(S_{g};k)$ and $MCG(S_{g}))$ are Braid group, Mapping class group (relative to $k$) and Mapping class group of orientable surface $S_{g}$, respectively. For $g\geq3,$ we have ...
King Khan's user avatar
  • 1,046
8 votes
1 answer
158 views

Generalisation of the Symmetric Group

For $m\in\mathbb{N}$, consider the group $G_m=\langle s_1,\dots,s_{n-1}\rangle$ generated by the relations \begin{align*} s_i^m&=1\\ s_is_j&=s_js_i &|i-j|>1 \\ s_is_js_i&=s_js_is_j &...
GossipM's user avatar
  • 405
2 votes
2 answers
223 views

Detect a knot from its fundamental group

I'm studying the braid closure and I ended up with the knot $K= (\sigma_1\,\sigma_2\,\sigma_1\,\sigma_2\,\sigma_1)_*$, here the notation is according Murasugi but it does not really matter. By using a ...
Giacomo Bascapè's user avatar
0 votes
1 answer
56 views

Prove or disprove $\langle a,b,c\rangle$ is free

Suppose that $\langle a,b\rangle$, $\langle b,c\rangle$, and $\langle a,c\rangle$ are free, is it true then that $\langle a,b,c\rangle$ is a free group of rank 3? We assume that $a,b,c$ are distinct ...
Sid Caroline's user avatar
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3 votes
1 answer
59 views

Every braid is a product of braid transpositions?

Let's start with a definition of an $n$-braid, as I understand them: Definition: Let $n$ be a natural number. In $ \mathbb{R}^2$, consider two collections of $n$ nodes $\{P_i\}$ and $\{Q_i\}$. Let ...
Noah Jensen's user avatar
1 vote
0 answers
186 views

Existence of composition series of infinite groups (especially the braid groups and the general linear group)

How can I argue that there are no composition series for the braid groups or the general linear group $GL(n,\mathbb{R})$ ? I have already learned that this is related to the infinite abelianization of ...
beta28's user avatar
  • 11
3 votes
1 answer
193 views

Symmetric Group $S_n$ and Artin Braid Group $B_n$: Differences in definition of composition?

I have been starting to look more closely at braid groups recently and I am slightly confused on the notion of "composition" in $S_n$ versus $B_n$. Function composition in $S_n$ is the usual,...
Noah Jensen's user avatar
2 votes
1 answer
368 views

The 3-component hopf link covering the trefoil

Let $H$ be the complement of the 3-component Hopf link, which is homeomorphic to the complement of the 3-chain link if you prefer, and let $T$ be the trefoil knot complement. I recently encountered ...
Ethan Dlugie's user avatar
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4 votes
0 answers
82 views

Are the central quotients of braid groups non-trivial free products?

The braid group $B_3$ has the property that its central quotient (i.e., $B_3 / Z(B_3)$) is isomorphic to the modular group $\mathrm{PSL}(2,\mathbb{Z})$. The modular group is known to be isomorphic to ...
James E Hanson's user avatar
3 votes
1 answer
274 views

Word problem in the braid group

The geodesic length of elements can be defined as the length of a minimal path from 1 to w in the Cayley graph of G. This length is dependent on the particular generating set X. The Word Problem is ...
Single Malt's user avatar
1 vote
0 answers
15 views

Are Braid groups special cases of lattice endomorphisms?

Considering that Braid groups are defined over linear 1D lattices, I am pondering what would be the multidimensional equivalent of Braid groups, when we consider endomorphisms between 2D lattices, or ...
lurscher's user avatar
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0 votes
0 answers
122 views

Easy to use program to calculate the HOMFLY polynomial of a braid word?

Easy to use program to calculate the HOMFLY polynomial of a braid word? Looking for a reference. Thanks!
user avatar
2 votes
1 answer
149 views

Calculating the faithful representation of the braid group

The braid group has the following presentation $B_n:\{s_1,...s_{n-1} : s_is_j = s_js_i \text{ for } |i-j|>1, s_is_{i+1}s_i=s_{i+1}s_is_{i+1} \}$ It was shown at some point that this group is linear,...
user avatar
3 votes
1 answer
55 views

Cardinality of the equivalence class of a given braid word?

The symmetric group $S_n$ has presentation $$S_n = \{ s_1,...,s_{n-1} | s_is_{i+1}s_i=s_{i+1}s_is_{i+1}, \text{ } s_i^2=1 , \text{ and } s_is_j = s_js_i \text{ for } |i-j| \geq 2\}$$ If we take away ...
user avatar
1 vote
2 answers
170 views

A family of groups as a monoidal category

1.Context My lecture notes present the following example of a monoidal category: Let $G:=(G_n)_{n\in \mathbb {N_0}}$ be a family of groups with $G_0$ the trivial group with one element. We define a ...
Max Demirdilek's user avatar
8 votes
1 answer
462 views

Is the braid group hyperbolic?

The braid groups satisfy a number of properties that one would expect of a hyperbolic group, liking having a solvable word problem, and having exponential growth. Are the braid groups hyperbolic ...
Quizzical's user avatar
  • 395
2 votes
0 answers
51 views

Examples of braided vector spaces

The following example of a braided vector space is given in my lecture notes: Let $k$ be a field, and $n>1$. Let $V$ be an $n$-dimensional vector space with ordered basis $(e_1, e_2,..., e_n)$. ...
Max Demirdilek's user avatar
3 votes
0 answers
214 views

Braided vector space: motivation?

In one of my courses we were given the definition of a braided vector space: Let $k$ be a field. A braided vector space is a pair $(V,s)$ with $V$ a $k$-vector space $s:V \otimes V \rightarrow V \...
Max Demirdilek's user avatar
6 votes
0 answers
126 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 ...
Levi's user avatar
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3 votes
1 answer
152 views

Isomorphism between braid groups

Let M be a connected topological manifold of dimension $\geq2$ and let $M^n=M\times\dots\times M$ be the product on $n\geq1$ copies of M with the product topology. Set $\mathcal{F}_n(M)=\{(u_1,u_2,\...
バレリオ's user avatar
2 votes
1 answer
135 views

A Tensor Calculation with Braids

I am trying to follow the derivation of the Jone's Polynomial from a braid representation presented in chapter 2 of Ohtsuki's Quantum Invariants. The representation of the braid $b$ with $n$ strands ...
Saud Molaib's user avatar
  • 1,058
6 votes
0 answers
301 views

Is this the Cayley graph of the braid group on three strands?

I have been attempting to draw the Cayley graph of the braid group $$ B_3 = \langle a, b \mid aba=bab \rangle$$ and I obtained something that almost seems too good to be true; here is a picture. This ...
Levi's user avatar
  • 4,786
2 votes
0 answers
60 views

Canonical length of a braid

Let $A=\Delta^m A_1 A_2...A_k$ be the Garside's normal form of a braid. Then its canonical length is $k$. I need to prove that for any $A,B \in B_n$ we have ${\rm len}(AB) \leq{\rm len}(A)+{\rm len}(B)...
user avatar
4 votes
2 answers
264 views

Finding all quotients of the braid group $B_5$ up to order $720$

Original Question. I am trying to find all quotients of the braid group $B_5$ on five strands up to order $6!=720$. From previous very slow implementations I found that there are not many. Some ...
Levi's user avatar
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