Knot group

Knot group
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "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,

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

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

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 has 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.

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