Torus Knot

A -torus knot is obtained by looping a string through the hole of a torus times with revolutions before joining its ends, where and are relatively prime. A -torus knot is equivalent to a -torus knot. All torus knots are prime (Hoste et al. 1998, Burde and Zieschang 2002). Torus knots are all chiral, invertible, and have symmetry group (Schreier 1924, Hoste et al. 1998).

Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are torus knots using the function KnotData[knot, "Torus"].

The link crossing number of a -torus knot is

(1)

(Williams 1988, Murasugi and Przytycki 1989, Murasugi 1991, Hoste et al. 1998). The unknotting number of a -torus knot is

(2)

(Adams 1991).

The numbers of torus knots with crossings are 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, ... (OEIS A051764). Torus knots with fewer than 11 crossings are summarized in the following table (Adams et al. 1991) and the first few are illustrated above.

knot	name
	trefoil knot	(3, 2)
	Solomon's seal knot	(5, 2)
		(7, 2)
		(4, 3)
		(9, 2)
		(5, 3)

The torus knot indices corresponding to knots on 16 or fewer crossings are , , , , , , , , , , , and (Hoste et al. 1998).

The , , and -torus knots are almost alternating knots (Adams 1994, p. 142).

The Jones polynomial of an -torus knot is

(3)

The bracket polynomial for the torus knot is given by the recurrence relation

(4)

where

(5)

The knot group of the -torus knot is

(6)

(Rolfsen 1976, p. 53).