Torus Knot - Properties

Properties

A torus knot is trivial iff either p or q is equal to 1 or −1.

Each torus knot is prime and chiral.

The (p,q) torus knot is equivalent to the (q,p) torus knot. This can be proved by moving the strands on the surface of the torus, which is nicely illustrated here. The (p,−q) torus knot is the obverse (mirror image) of the (p,q) torus knot. The (−p,−q) torus knot is equivalent to the (p,q) torus knot except for the reversed orientation.

Any (p,q)-torus knot can be made from a closed braid with p strands. The appropriate braid word is

(This formula assumes the common convention that braid generators are right twists, which is not followed by the Wikipedia page on braids.)

The crossing number of a (p,q) torus knot with p,q > 0 is given by

c = min((p−1)q, (q−1)p).

The genus of a torus knot with p,q > 0 is

The Alexander polynomial of a torus knot is

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

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

Torus knots are the only knots whose knot groups have nontrivial center (which is infinite cyclic, generated by the element in the presentation above).