Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in .
Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between certain pairs of inequivalent knots. This is because an equivalence between two knots is a self-homeomorphism of that is isotopic to the identity and sends the first knot onto the second. Such a homeomorphism restricts onto a homeomorphism of the complements of the knots, and this restricted homeomorphism induces an isomorphism of fundamental groups. However, it is possible for two inequivalent knots to have isomorphic knot groups (see below for an example).
The knot group (or fundamental group of an oriented link in general) can be computed in the Wirtinger presentation by a relatively simple algorithm.
- The unknot has knot group isomorphic to Z.
- The trefoil knot has knot group isomorphic to the braid group B3. This group has the presentation
- A (p,q)-torus knot has knot group with presentation
- The figure eight knot has knot group with presentation
- The square knot and the granny knot have isomorphic knot groups, yet these two knots are inequivalent.
- Hazewinkel, Michiel, ed. (2001), "Knot and Link Groups", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104