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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

<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 on 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.

[edit] 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>

Torus knots are the only knots whose knot groups have non-trivial center (which is infinite cyclic, generated by the element <math>x^p = y^q</math> in the presentation above.

[edit] External links

• Weisstein, Eric W., Torus Knot at MathWorld.zh:环面扭结