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 positive integers greater than 1. 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 crossings 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)

(Kronheimer and Mrowka 1993, Adams 1994).

The numbers of torus knots with , 2, ... crossings, identifying each knot with its mirror image, 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 -torus knots are alternating. The - and -torus knots are almost alternating links (Adams 1994, p. 142; Abe 2009, p. 363).

The Jones polynomial of an -torus knot, where and are relatively prime positive integers greater than 1, is

(3)

For a positive integer , let be the standard closure diagram of the two-strand braid with positive crossings. It represents the -torus knot when is odd and a two-component link when is even. For , its unnormalized bracket polynomial satisfies the recurrence relation

(4)

where

(5)

Here, is an unknot diagram with a positive curl.

The knot group of the -torus knot is

(6)

(Rolfsen 1976, p. 53).