Recently, I came across a subgroup of the braid group $B_{2n}$ that I'm calling the "loop manipulation" group $H_n$.
The idea is that we treat pairs of adjacent strands in the braid group as "loops" $i = 1, \dots, n$.
It is generated by the following elements: 
+ $T_i$ for $i = 1, \dots, n$, 
+ $C_i$ for $i = 1, \dots, n-1$
+ $P_i$ for $i = 1, \dots, n-1$.

The nomenclature is meant to suggest that 
$T_i$ is a "twist", $C_i$ is a
"cross", and $P_i$ is a "pass".
Here is an illustration of the generators: [The loop manipulation generators][1]. (Also, there is a copy on my website [here](https://pgadey.ca/office-camera/2024-04-01t092651-0400/).) Generally, $H_n$ has a lot in common with the braid group $B_n$.
The $C_i$ generators are analogous to the the usual generators $\sigma_i$ of the braid group $B_{n}$ so we have:

1. The usual braid relation: $C_i C_{i+1} C_i = C_{i+1}C_iC_{i+1}$ for $i = 1, \dots, n-1$.
2. The usual braid commutation relation: If $|i - j| \geq 2$ then $C_iC_j = C_jC_i$

Playing with lots of little diagrams has convinced me of the following relations:

3. $T_iT_j = T_jT_i$ for all $i$ and $j$.
4. $T_iC_i = C_iT_{i+1}$ for $i = 1, \dots, n-1$.
5. $T_iP_i = P_iT_{i+1}$ for $i = 1, \dots, n-1$.

I've got two questions about $H_n$:

1. Do the relations above determine $H_n$ uniquely? To put it formally: If an abstract group
   generated by $T_i$, $C_i$, and $P_i$ satisfied these relations, would it be
   isomorphic to $H_n$?
2. Does this group appear in the literature anywhere? 


  [1]: https://i.sstatic.net/h7RBD.jpg