Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to 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 $P_n$ defined by $$A_{j,k}=(\sigma_{k-1}\sigma_{k-2}\cdots\sigma_{j+1})\sigma_j^2(\sigma_{j+1}^{-1}\cdots\sigma_{k-2}^{-1}\sigma_{k-1}^{-1})$$ and $g$ is an element of the subgroup of $P_n$ generated by $A_{1,k},A_{2,k},\cdots,A_{k-1,k}$.

Exercise 3.4 of the book says that the Relations 3 above can be replaced by

$A_{j,k}$ commutes with $gA_{j,k}g^{-1}$ where $g$ is an element of the subgroup of $P_n$ generated by $A_{j,j+1},A_{j,j+2},\cdots,A_{j,n}$.

I am wondering what if we replace the Relation 3 by

$A_{j,k}$ commutes with $gA_{j,k}g^{-1}$ where $g$ is an element of $P_n$.

Is it still the same definition as the original one? If no, that means that there exists a pure braid $g$ such that $A_{j,k}$ doe not commute with $gA_{j,k}g^{-1}$. Anyone know such an example?

**Edit**: My last paragraph was not clear. It should be that "there exists a pure homotopy braid $g$ such that $A_{j,k}$ (regarded as a homotopy braid) doe not commute with $gA_{j,k}g^{-1}$ in $\tilde{P_n}$".