Torus Knot

A $(p,q)$-torus Knot is obtained by looping a string through the Hole of a Torus $p$ times with $q$ revolutions before joining its ends, where $p$ and $q$ are Relatively Prime. A $(p,q)$-torus knot is equivalent to a $(q,p)$-torus knot. The Crossing Number of a $(p,q)$-torus knot is

$$c=\min\{p(q-1), q(p-1)\}$$ (1)

(Murasugi 1991). The Unknotting Number of a $(p,q)$-torus knot is

$$u={\textstyle{1\over 2}}(p-1)(q-1)$$ (2)

(Adams 1991).

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 $(2,q)$, $(3,4)$, and $(3,5)$-torus knots are Almost Alternating Knots.

The Jones Polynomial of an $(m,n)$-Torus Knot is

$${t^{(m-1)(n-1)/2}(1-t^{m+1}-t^{n+1}+t^{m+n})\over 1-t^2}.$$ (3)

The Bracket Polynomial for the torus knot $K_n=(2,n)$ is given by the Recurrence Relation

$$\left\langle{K_n}\right\rangle{}=A\left\langle{K_{n-1}}\right\rangle{}+(-1)^{n-1}A^{-3n+2},$$ (4)

where

$$\left\langle{K_1}\right\rangle{}=-A^3.$$ (5)

See also Almost Alternating Knot, Hyperbolic Knot, Knot, Satellite Knot, Solomon's Seal Knot, Trefoil Knot

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.