Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to self-intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.

I am reading *A study of Braids* by K. Murasugi and B. I. Kurpita. In Chapter 7, Theorem 3.3, a presentation of $\tilde{B_n}$ is given as follows:

Generators: $\sigma_1,\sigma_2,\cdots, \sigma_{n-1}$

Relations:

$\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$;

$\sigma_i\sigma_j=\sigma_j\sigma_i$ for $|i-j|\geq 2$;

$A_{j,k}$ commutes with $gA_{j,k}g^{-1}$, where $A_{j,k}$ are the usual generators of the pure braid group $P_n$ and $g$ is an element of the subgroup of $P_n$ generated by $A_{1,k},A_{2,k},\cdots,A_{k-1,k}$.

My main question is:

**What is a presentation of the homotopy pure braid group $\tilde{P_n}$? Is it the usual presentation of the pure braid group $P_n$ plus relation 3 above?**

Another question:

The authors of the book said that "it is quite possible that $\tilde{B_n}$ may not be finitely presented". The book was written around 15 years ago; is there any new result proving if $\tilde{B_n}$ is finitely presented?