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: 


1. $\sigma\_i\sigma\_{i+1}\sigma\_i=\sigma\_{i+1}\sigma\_i\sigma\_{i+1}$;
2. $\sigma\_i\sigma\_j=\sigma\_j\sigma\_i$ for $|i-j|\geq 2$;
3. $A\_{j,k}$ 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 $\tilde{P\_n}$".



Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups?
Consider $A\_{2,3}=\sigma\_2^2$. 
We have that $\sigma\_2^2$ does not commute with $\sigma\_1^2.\sigma\_2^2.(\sigma\_1^2)^{-1}$ in the homotopy pure braid group, i.e, $[\sigma\_1^2,\sigma\_2^2]$ is non trivial. 


[Habegger and Lin](http://math.ucr.edu/~xl/papers/habegger-lin.pdf) showed that the homotopy pure braid group is classified by Milnor invariants without repeated indices (more precisely, they classified string links up to homotopy, and showed that any string link is homotopic to a pure braid).
We have that Milnor invariant $\mu\_{123}$ detects the commutator $[\sigma\_1^2,\sigma\_2^2]$. 
Notice that the closure of $[\sigma\_1^2,\sigma\_2^2]$ is isotopic to Borromean rings, which is typically detected by Milnor link invariants $\overline{\mu}\_{123}$; see Milnor's [paper](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=AUCN&pg7=RVCN&pg8=ET&review_format=html&s4=Milnor&s5=Isotopy%20of%20links&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=92150).


