# Questions tagged [braid-groups]

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### Do there exist any variational principles on the space of braids (or knots)?

This is very speculative question and I do not know where to start looking up the literature, or if what I am looking for is even mathematically possible/meaningful. Q: I am interested in finding out ...
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### Permuting $n$ points in a $2$-manifold

Given $n$ points on a connected $2$-manifold $M$, I'd like to consider the homotopy classes of paths that "permute" these points. Edit (Clarifying what I mean by this): Given a set of $n$ distinct ...
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### Software for finding conjugates in the Braid group

The conjugacy problem for the braid group was solved by Garside, and gives an algorithm for determining whether two braids are conjugate. Since this algorithm is rather tedious, I was wondering if ...
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### Finite quotients of surface braid groups

Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$. There is a braid $A_{12} \in \Sigma_b$ ...
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### When do elements in the braid group $B_n$ commute?

I have been looking around for an answer to this question, but I have not been able to find anything. My question is: Is it known how to tell whether two elements $b_1, b_2 \in B_n$ commute? EDIT: ...
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### Is there an analog of Reidemeister's theorem for braids in a surface?

Reidemeister's classical theorem describes the set of links in $\mathbb R^3$ up to isotopy as the set $\{ \textrm{diagrams in } \mathbb R^2\textrm{ with crossings}\}$ modulo certain local relations on ...
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### Is the word problem in the braid group quotient $B_{n}/N$ solvable where $N$ is the normal subgroup generated by conjugates of $\sigma_{i}^{2r}$?

Let $r\geq 2$. Let $N$ be the normal subgroup of $B_{n}$ generated by conjugates of $\sigma_{i}^{2r}$. Then is the word problem in the quotient group $B_{n}/N$ solvable (in polynomial time)? ...
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### How quickly can one compute the Hurwitz action of braid groups on finite groups?

Let $G$ be a finite group. Define the Hurwitz action of $B_{n}$ on $G^{n}$ by letting $(x_{1},...,x_{n})\sigma_{i}=(x_{1},...,x_{i}x_{i+1}x_{i}^{-1},x_{i},x_{i+2},...,x_{n})$. I wonder what algorithms ...
### Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal. Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
A braid element $b$ in the braid Group $B_n$ is an essential braid if it does not have at least one of the generators in it. For example, $b=\sigma_1\sigma_2\sigma_4 \in B_5$ is essential since \$\...