Knot group

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,

$\pi _{1}(\mathbb {R} ^{3}-K).$

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

Examples

• The unknot has a knot group isomorphic to Z, the infinite cyclic group.
• The trefoil knot has a knot group isomorphic to the braid group B₃. This group has the presentation $\langle x,y\mid x^{2}=y^{3}\rangle$ or $\langle a,b\mid aba=bab\rangle$.
• A (p,q)-torus knot has knot group with presentation $\langle x,y\mid x^{p}=y^{q}\rangle$.