Torus knot

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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 q times around a circle inside the torus, which goes all the way around the torus, and p 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 $(r - 2) 2 + z 2 = 1$ (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.

Each torus knot is prime and chiral. 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 complement of a torus knot in the 3-sphere is a Seifert-fibered manifold, fibred over the disc with two singular fibres.

Let Y be the p-fold dunce cap with a disk removed from the interior, Z be the q-fold dunce cap with a disk removed its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle. The knot complement of the (p, q)-torus knot deformation retracts to the space X. Therefore, 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 $x p = y q$ in the presentation above).