Knot group

Definition
The knot group of a knot $$K$$ (where $$K$$ is a homeomorphic copy of $$S^1$$ inside $$S^3$$) is the fundamental group of the knot complement, viz., $$\pi_1(S^3 \setminus K)$$. Note that $$S^3 \setminus K$$ is path-connected, and its first homology group is $$\mathbb{Z}$$. Thus the Abelianization of the knot group is $$\mathbb{Z}$$.