# Braid group and knot group

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 $S^{n+2}$?

• The first question is answered (negatively) here: math.stackexchange.com/questions/2118701/… – Igor Rivin Aug 8 '17 at 2:50
• To summarize Igor's link, for $n\ge 4$, $B_n$ contains a copy of $\mathbf{Z}^3$ (use $\sigma_1$, $\sigma_3$, and a generator of the center), while an aspherical 3-manifold group can't contain $\mathbf{Z}^3$ unless it's virtually abelian. – YCor Aug 8 '17 at 15:07

This is an incomplete answer to the last question, but perhaps someone can supply the last piece of information to complete it. The question of whether a finitely presented group G is a higher-dimensional knot group was resolved by Kervaire. Necessary and sufficient conditions are (1) G is the normal closure of one element, (2) $H_1(G) \cong {\mathbb Z}$, and (3) $H_2(G) = 0$.
The first two of these hold for the braid groups. I'm sure that the calculation of $H_2(B_n)$ is somewhere in the literature, but I couldn't locate it.