Mathematics Stack Exchange
Mathematics Stack Exchange

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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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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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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$\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ 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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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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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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Braid groups as knot groups?

Let $B_n$ denote the braid group on $n$ strands. The knot group of the trefoil knot is isomorphic to $B_3$. Are there other knots $K_n$ such that the knot group of $K_n$ is $B_n$?
user101010
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Residually Finite Braid Group

In Braid Groups of Kassel, Turaev, it mentions that $\mathcal{B}_n$ is a residually finite group. The definition that they give as a residually finite group is a group $G$ such that for each $g\in G-\{e_G\}$ ($e_G$ the identity of $G$), there exists…
iam_agf
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Equivalence of two relations in Braid groups

Let $B_n$ be the braid group; that is, a group generated by $\sigma_1,\cdots,\sigma_{n-1}$ with relations $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ for $i=1,\cdots,n-2$; $\sigma_i\sigma_j=\sigma_j\sigma_i$ if…
Zuriel
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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 Cayley diagram for $n=3$, which turns out to have…
SAB
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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 a short exact sequence $$1\longrightarrow…
King Khan
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