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,

$\pi_1(\mathbb{R}^3 \backslash K).$

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 S'³

[edit] Properties

Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between inequivalent knots. This is because an equivalence between two knots is a self homeomorphism of $\mathbb{R}^3$ isotopic to the identity sending the first knot onto the second. Clearly the homeomorphism restricts onto a homeomorphism of the complement of the knot therefore inducing an isomorphism in the fundamental group of the knot complement. However, two knots can have isomorphic knot groups without necessarily being equivalent (see below for an example).

The abelianization of a knot group is always isomorphic to the infinite cyclic group Z; this follows because the abeliazation agrees with the first homology group, which can be easily computed.

The knot group (or fundamental group of an oriented link in general) can be computed in the Wirtinger presentation by a relatively simple algorithm.

[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 $\langle x,y \mid x^2 = y^3 \rangle$ or $\langle a, b \mid aba = bab \rangle$ .
A (p,q)-torus knot has knot group with presentation $\langle x,y \mid x^p = y^q \rangle$ .
The figure eight knot has knot group with presentation $\langle x,y \mid x^{-1}yxy^{-1}xy=yx^{-1}yx\rangle$
The square knot and the granny knot have isomorphic knot groups, yet these two knots are inequivalent.