Kervaire's Characterization Theorem

Let $G$ be a Group, then there exists a piecewise linear Knot $K^{n-2}$ in $\Bbb{S}^n$ for $n\geq 5$ with $G=\pi_1(\Bbb{S}^n-K)$ Iff $G$ satisfies

1. $G$ is finitely presentable,

2. The Abelianization of $G$ is infinite cyclic,

3. The normal closure of some single element is all of $G$,

4. $H_2(G)=0$; the second homology of the group is trivial.

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 350-351, 1976.