Torus knot

From Wikipedia, the free encyclopedia

You have new messages (last change).
A (3,7)-3D torus knot rendered by Apple Grapher.
A (3,7)-3D torus knot rendered by Apple Grapher.

In knot theory, a torus knot is a special kind of knot which lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. The (p,q)-torus knot winds p times around a circle inside the torus, which goes all the way around the torus, and q times around a line through the hole in the torus, which passes once through the hole, (usually drawn as an axis of symmetry). If p and q are not relatively prime, then we have a torus link with more than one component.

The (p,q)-torus knot can be given by the parameterization

x = \left(2+\cos\left(\frac{q\phi}{p}\right)\right)\cos\phi
y = \left(2+\cos\left(\frac{q\phi}{p}\right)\right)\sin\phi
z = \sin\left(\frac{q\phi}{p}\right)

This lies on the surface of the torus given by (r − 2)2 + z2 = 1 (in cylindrical coordinates).

Torus knots are trivial iff either p or q is equal to 1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.

[edit] Properties

Diagram of a (3,8)-torus knot.
Diagram of a (3,8)-torus knot.

Each torus knot is prime and chiral. The complement of a torus knot in the 3-sphere is a Seifert-fibered manifold, fibred over the disc with two singular fibres. A (p,q)-torus knot is isotopic to a (r,s)-torus knot if and only if p / q = r / s or p / q = s / r. Any (p,q)-torus knot can be made from a closed braid with p strands. The appropriate braid word is

(\sigma_1\sigma_2\cdots\sigma_{p-1})^q.

The crossing number of a torus knot is given by

c = min((p−1)q, (q−1)p).

The genus of a torus knot is

g = \frac{1}{2}(p-1)(q-1).

The Jones polynomial of a (right-handed) torus knot is given by

t^{(p-1)(q-1)/2}\frac{1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^2}.

The knot group of a torus knot has the presentation

\langle x,y \mid x^p = y^q\rangle.

Torus knots are the only knots whose knot groups have non-trivial center (which is infinite cyclic, generated by the element xp = yq in the presentation above.

[edit] External links

Retrieved from "http://en.wikipedia.org../../../t/o/r/Torus_knot.html"
In other languages