Knot group

From Wikipedia, the free encyclopedia

In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R³,

Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between inequivalent knots.

The abelianization of a knot group is always isomorphic to the infinite cyclic group Z.

[edit] Examples

• 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 or .
• A (p,q)-torus knot has knot group with presentation .
• The square knot and the granny knot have isomorphic knot groups, yet these two knots are inequivalent.