It is well known that braid groups and knot groups share many common properties. For example, they have the same $H\_1$ and they are both residually finite and (hence) Hopfian. On the other hand, we know that $B\_2$ and $B\_3$ are isomorphic to the knot groups of unknot and trefoil knot respectively. My question is, for any $n\geq 4$, is there a knot with knot group $B\_n$? 


If not, how about high dimensional knot $S^n$ in $S^{n+2}$?



Braid group and knot group
This is a simple addendum to Danny's answer (because I can't use the citation thing in comments): Homology of braid groups has been computed by a number of people, and [an extensive survey](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.488.4621&rep=rep1&type=pdf) is:


*Vershinin, Vladimir V.*, [**Homology of braid groups and their generalizations**](https://eudml.org/doc/244559), Jones, Vaughan F. R. (ed.) et al., Knot theory. Proceedings of the mini-semester, Warsaw, Poland, July 13--August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42, 421-446 (1998). [ZBL0905.20032](https://zbmath.org/?q=an:0905.20032).


Unless I am misinterpreting the results, there is nontrivial homology in all degrees, so the answer to the OP's second question is negative also.


