Mathematics Stack Exchange
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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.

Algebraically a braid group $B_n$ is generated by elements $\sigma_1,\dotsc, \sigma_{n-1}$ subject to the relations $$ \sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} \;\text{ for }\; i \in \{1,2,\dotsc, n-2\} \quad\text{ and }\quad \sigma_i\sigma_j = \sigma_j\sigma_i \;\text{ for }\; |i-j|>2 \,. $$ Intuitively, you think of $B_n$ as the group of braidings of $n$ strands, the $\sigma_i$ representing simple crossings between adjacent strands. Seeing illustrations of this is incredibly helpful in understanding braid groups, so please check out Wikipedia for a more thorough exposition.

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Geometric way to view the truncated braid groups?

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. I also asked a related question on MO, although hopefully this question should be easier. There…
Dan Rust
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General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid group $B_n$, namely as the mapping class group of the…
anon
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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 groups (cyclic of prime order) are all quotients…
Ian Gershon Teixeira
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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 & |i-j|=1 \end{align*} If $m=1$, $G_m$ is trivial. If $m=2$, $G_m$…
GossipM
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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 groups? If not, is there any obvious property of…
Quizzical
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$\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ representation of $B_3$ braid group

I've been trying to find a representation of the braid group $B_3$ acting on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ but I can't find it anywhere. From what I understand I have to find two $8 \times 8$ matrices $\sigma_i$ satisfying…
GKiu
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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 once. For a braid group of about four strands it…
Single Malt
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Are any braid groups linear algebraic group?

$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}\DeclareMathOperator\ASL{ASL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\Decla…
Ian Gershon Teixeira
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Algorithm to determine if $A = B^k$ for any $k\geq 0$ if $A, B$ are special unitary matrices

I am working with the Lawrence-Krammer representation of $B_n$ and need to find a way to determine if, given any two matrices $A, B$ in the image of the representation, there exists $k\in\mathbb{Z}^+$ such that $A = B^k$. I am working with the…
Finite Matrix
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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 homomorphism $B_m \to B_n$. If $m>n$, there are surjective maps…
Levi
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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 might require some explanation: The vertices of…
Levi
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Explicit Formula for Cabling of Braids

Given the Artin braid groups on $n$ and $m$ strands $Br_n$ and $Br_m$, there are cabling operations $\circ_k:Br_n\times Br_m\to Br_{n+m-1}$ that take a braid $\beta\in Br_m$ and replace the $k$th strand of a braid in $Br_n$ with $\beta$. See the…
Jonathan Beardsley
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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 $S$ is neither the sphere nor the projective…
King Khan
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Geometric reason why conjugation by an element in $B_3$ inverts this element?

Let $B_3$ be the braid group on three strands. I was looking at an element in $B_3$, which I will write in the standard presentation: $$(\sigma_2\sigma_1\sigma_2)^{-1}\sigma_1^3\sigma_2^{-3}(\sigma_2\sigma_1\sigma_2)$$ and I was able to explicitly…
Andres Mejia
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Is there a way to get all the permutation braids of a braid group?

Geometrically, it's easy to "draw" the permutation braids, but I was wondering if there was an algorithm to write down all the permutation braids in terms of the Artin generators. I had a few ideas, but none of them seem quite feasible or…
quasicoherent_drunk
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