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).

REFERENCES:

Adams, C.; Hildebrand, M.; and Weeks, J. "Hyperbolic Invariants of Knots and Links." Trans. Amer. Math. Soc. 326, 1-56, 1991.

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994.

Burde, G. and Zieschang, H. Knots, 2nd rev. ed. Berlin: de Gruyter, 2002.

Gray, A. "Torus Knots." §9.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 209-215, 1997.

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots." Math. Intell. 20, 33-48, Fall 1998.

Kronheimer, F. B. and Mrowka, T. S. "Gauge Theory for Embedded Surfaces I." Topology 32, 773-826, 1993.

Kronheimer, F. B. and Mrowka, T. S. "Gauge Theory for Embedded Surfaces II." Topology 34, 37-97, 1995.

Murasugi, K. "On the Braid Index of Alternating Links." Trans. Amer. Math. Soc. 326, 237-260, 1991.

Murasugi, L. and Przytycki, J. "The Skein Polynomial of a Planar Star Product of Two Links." Math. Proc. Cambridge Philos. Soc. 106, 273-276, 1989.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976.

Schreier, O. "Über die Gruppen ." Abh. Math. Sem. Univ. Hamburg 3, 167-169, 1924.

Sloane, N. J. A. Sequence A051764 in "The On-Line Encyclopedia of Integer Sequences."

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 275-277, 1999.

Williams, R. F. "The Braid Index of an Algebraic Link." Braids (Santa Cruz, CA, 1986). Providence, RI: Amer. Math. Soc., 1988.