Torus knot

In knot theory, a torus knot is a special kind of knot which lies on the surface of a torus in R³. Each torus knot is specified by a pair of coprime integers p and q. The (p,q)-torus knot winds p times around one cycle of the torus and q times around the other.

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

<math>x = \left(2+\cos\left(\frac{q\phi}{p}\right)\right)\cos\phi<math>

<math>y = \left(2+\cos\left(\frac{q\phi}{p}\right)\right)\sin\phi<math>

<math>z = \sin\left(\frac{q\phi}{p}\right)<math>

This lies of the surface of the torus given by <math>(r-2)^2 + z^2 = 1<math> (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.

Properties

Each torus knot is prime and chiral. The (p,q)-torus knot is equivalent to the (q,p)-torus knot. Any (p,q)-torus knot can be made from a closed braid with p strands. The appropriate braid word is

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

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

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

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

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

The knot group of a torus knot has the presentation

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

External links

• Eric W. Weisstein, Torus Knot at MathWorld.