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. The Crossing Number of a -torus knot is

(1) |

(2) |

Torus knots with fewer than 11 crossings are the Trefoil Knot 03-001
(3, 2), Solomon's Seal Knot
05-001
(5, 2), 07-001
(7, 2), 08-019
(4, 3), 09-001
(9, 2), and 10-124
(5, 3) (Adams *et al. *1991). The only Knots which are not Hyperbolic Knots are
torus knots and Satellite Knots (including Composite Knots). The
, , and -torus knots are Almost Alternating Knots.

The Jones Polynomial of an -Torus Knot is

(3) |

(4) |

(5) |

**References**

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

Gray, A. ``Torus Knots.'' §8.2 in *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 155-161, 1993.

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

© 1996-9

1999-05-26