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