<p>The following notion of 'tameness' seems interesting. Say that a braid $\beta \in B_n$ is <em>tamed</em> if there is a set of $n$ rectilinear curves $\cal{C}$ over the 3-dimensional cylinder $S_1 \times [0,n]$ such that when projecting $\cal{C}$ on the $x$ plane, we obtain the graphic representation of $\beta$. Clearly, this notion is closed by inversion and composition, and we can therefore define the notion of <em>tamed braid group</em> (TBG). This leads to the following definition: a pair $(G,H)$ of braid groups is a <em>tamed pair</em> if the shuffle product $G \| H$ is a TBG.</p>

<p>Let us precise this notation. Given a braid $\beta \in B_n$, and given a $k$-subset $I \subseteq B_n$, we denote by $\beta | I$ the braid in $B_k$ obtained by keeping the strands of $\beta$ starting at an index of $I$. Now, if $G \subseteq B_m$ and $H \subseteq B_n$, we define $G \| H$ is the subset of $B_{m+n}$ containing the braids $\beta$ such that $\beta | I \in G, \beta | J \in H$ for some partition $(I,J)$ of $[n+m]$.</p>

<p>This definition seems to have some interesting properties, in particular $G$ and $H$ play symmetric roles. There might be an alternate way to define 'tameness' though, but this is currently the only one I can think of. Note that there is a different notion of <em>tame/wild knot</em> in knot theory, but it doesn't seem very relevant to us as it only makes sense for infinite knots.</p>

CC BY-SA 3.0
2014-04-15T01:58:11.140Z
2014-04-15T01:58:11.140Z
Super8
<p>I was looking for interesting questions pertaining to braid theory. I don't know if the following are considered important, but I'd like to ask:</p>

<p>(1) in relation with the following <a href="https://cstheory.stackexchange.com/questions/21936/extending-the-notion-of-independence">link</a>, is it true that every subgroup of $B_n$ has a generating set of size at most $f(n)$, for some function $f$? This would imply that $B_n$ is a Noetherian group.</p>

<p>(2) are there positive algorithmic results for braid groups described by generators?</p>

<p>(3) it is known that every finite group of order $n$ can be embedded in $S_n$, so it is possible to define a notion of 'pseudo-order $n$' for an infinite group that allows embedding in $B_n$?</p>

<p>(4) are there interesting partial orders to study on the braids of fixed size? Dehornoy showed that $B_n$ admits a left-invariant total order $\leq$, but I'm looking for a partial order $\preceq$ that would have a more 'topological' flavor (possibly such that $\preceq \subseteq \leq$).</p>

<p>This list is certainly incomplete, so please feel free to point out other questions in the field that are deemed important.</p>

CC BY-SA 3.0
2014-04-12T14:06:36.937Z
Only the second question is on-topic here. You should ask the other three (separately!) at MathOverflow.
Jeffε
CC BY-SA 3.0
2014-04-12T15:44:26.083Z
Much more suitable for mathoverflow.net
Joshua Grochow
CC BY-SA 3.0
2014-04-18T22:41:59.550Z
re the comments pov off topic, there is some mostly-sketchy-conjectured link between group generator growth and complexity theory eg [gromovs thm](http://rjlipton.wordpress.com/2013/06/20/three-theorems-about-growth/) etc., but agreed that on the face of it this sounds like pure group theory & it would be better to edit to show more connection/motivation wrt TCS
vzn
CC BY-SA 4.0
2020-01-09T09:46:03.103Z
A nice list of unsolved  problems for braid groups (and more generally groups) is the Kourovka notebook https://kourovkanotebookorg.files.wordpress.com/2019/12/19tkt-2.pdf
For instance "Is the conjugacy problem in the braid group $B_n$ in the class NP?"
thibo
<gr.group-theory><algebraic-topology>
Some questions (still) unresolved about braids?
NisaiVloot
# Some questions (still) unresolved about braids?

(Asked by: username_0 on 2014-04-10)

I was looking for interesting questions pertaining to braid theory. I don't know if the following are considered important, but I'd like to ask:

(1) in relation with the following [link](https://cstheory.stackexchange.com/questions/21936/extending-the-notion-of-independence), is it true that every subgroup of $B\_n$ has a generating set of size at most $f(n)$, for some function $f$? This would imply that $B\_n$ is a Noetherian group.

(2) are there positive algorithmic results for braid groups described by generators?

(3) it is known that every finite group of order $n$ can be embedded in $S\_n$, so it is possible to define a notion of 'pseudo-order $n$' for an infinite group that allows embedding in $B\_n$?

(4) are there interesting partial orders to study on the braids of fixed size? Dehornoy showed that $B\_n$ admits a left-invariant total order $\leq$, but I'm looking for a partial order $\preceq$ that would have a more 'topological' flavor (possibly such that $\preceq \subseteq \leq$).

This list is certainly incomplete, so please feel free to point out other questions in the field that are deemed important.

**Tags:** gr.group-theory, algebraic-topology

> **Comments:**
> **username_1**: Only the second question is on-topic here. You should ask the other three (separately!) at MathOverflow.  
> **username_2**: Much more suitable for mathoverflow.net  
> **username_3**: re the comments pov off topic, there is some mostly-sketchy-conjectured link between group generator growth and complexity theory eg [gromovs thm](http://rjlipton.wordpress.com/2013/06/20/three-theorems-about-growth/) etc., but agreed that on the face of it this sounds like pure group theory & it would be better to edit to show more connection/motivation wrt TCS  
> **username_4**: A nice list of unsolved  problems for braid groups (and more generally groups) is the Kourovka notebook https://kourovkanotebookorg.files.wordpress.com/2019/12/19tkt-2.pdf For instance "Is the conjugacy problem in the braid group $B_n$ in the class NP?"  

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## 1 answers

### Answer by username_5 on 2014-04-15. Score: 0

The following notion of 'tameness' seems interesting. Say that a braid $\beta \in B\_n$ is *tamed* if there is a set of $n$ rectilinear curves $\cal{C}$ over the 3-dimensional cylinder $S\_1 \times [0,n]$ such that when projecting $\cal{C}$ on the $x$ plane, we obtain the graphic representation of $\beta$. Clearly, this notion is closed by inversion and composition, and we can therefore define the notion of *tamed braid group* (TBG). This leads to the following definition: a pair $(G,H)$ of braid groups is a *tamed pair* if the shuffle product $G \| H$ is a TBG.

Let us precise this notation. Given a braid $\beta \in B\_n$, and given a $k$-subset $I \subseteq B\_n$, we denote by $\beta | I$ the braid in $B\_k$ obtained by keeping the strands of $\beta$ starting at an index of $I$. Now, if $G \subseteq B\_m$ and $H \subseteq B\_n$, we define $G \| H$ is the subset of $B\_{m+n}$ containing the braids $\beta$ such that $\beta | I \in G, \beta | J \in H$ for some partition $(I,J)$ of $[n+m]$.

This definition seems to have some interesting properties, in particular $G$ and $H$ play symmetric roles. There might be an alternate way to define 'tameness' though, but this is currently the only one I can think of. Note that there is a different notion of *tame/wild knot* in knot theory, but it doesn't seem very relevant to us as it only makes sense for infinite knots.


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