For a knot $K \subset S^3$, the knot group is the fundamental group of the complement: $\pi_K = \pi_1(S^3 \setminus K)$. Since $S^3 \setminus K$ is an open 3-manifold with the homotopy type of a compact manifold with boundary (the knot exterior $E_K = S^3 \setminus N(K)$ where $N(K)$ is an open tubular neighborhood), the knot group is finitely presented. For a link $L = K_1 \cup \cdots \cup K_\mu$, the link group $\pi_1(S^3\setminus L)$ similarly captures the linking information.

Given an oriented knot diagram $D$ with arcs $x_1,\ldots,x_n$ (segments between consecutive undercrossings) and crossings $c_1,\ldots,c_n$, the Wirtinger presentation of the knot group is: $\pi_K = \langle x_1,\ldots,x_n \mid r_1,\ldots,r_n\rangle$ where at each crossing $c_k$ where arc $x_i$ passes over and arcs $x_j,x_\ell$ are the under-strands: $r_k: x_\ell = x_i x_j x_i^{-1}$ (positive crossing) or $r_k: x_\ell = x_i^{-1}x_j x_i$ (negative crossing). Any one relation is a consequence of the others, so one may be omitted.

The peripheral subgroup of the knot group is the image of $\pi_1(\partial N(K)) \cong \mathbb{Z}\oplus\mathbb{Z}$ in $\pi_K$, generated by the meridian $\mu$ (a small loop linking $K$ once) and the longitude $\lambda$ (a curve on $\partial N(K)$ parallel to $K$ with zero linking number). The pair $(\mu,\lambda)$ is defined up to conjugacy. Waldhausen's theorem: the knot group together with its peripheral structure determines the knot type (for prime knots). The group alone does not suffice -- there exist distinct knots with isomorphic knot groups (e.g., square and granny knots).