Knot group

This article describes a knot invariant

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 $\mathbb{Z}$. Further information: homology of knot complement