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 $i,j\in\{1,\cdots,n-1\}$ and $|i-j|\geq 2$.
For $1\leq i<j\leq n$, let the usual generator $A_{i,j}$ of pure braid groups be defined as $$A_{i,j}=(\sigma_{j-1}\sigma_{j-2}\cdots\sigma_{i+1})\sigma_i^2(\sigma_{i+1}^{-1}\cdots\sigma_{j-2}^{-1}\sigma_{j-1}^{-1}).$$
My question: Is there a formula for the commutator $[A_{p,q},A_{s,t}]$?
By the way, I found the following related result in Artin braid groups and the homotopy groups by J. Li and J. Wu:
