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 toa -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 Knot05-001(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 aretorus 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.

• Gram-Charlier Series
• Gram Determinant
• Gram Matrix
• Gram-Schmidt Orthonormalization
• Gram Series
• Gram's Inequality
• Granny Knot
• Graph (Function)
• Graph (Graph Theory)
• Graphical Partition
• Graph Theory
• Graph Two-Coloring
• Grassmann Algebra
• Grassmann Coordinates
• Grassmann Manifold
• Gray Code
• Great Circle
• Great Cubicuboctahedron
• Great Deltoidal Hexecontahedron
• Great Deltoidal Icositetrahedron
• Great Dirhombicosidodecacron
• Great Dirhombicosidodecahedron
• Great Disdyakis Dodecahedron
• Great Disdyakis Triacontahedron
• Great Ditrigonal Dodecacronic Hexecontahedron