TorusKnot

The torus knot $T(p,q)$, winding $p$ times one way and $q$ the other round a torus — a value carrying both a closed form and, when a word for it exists, the TorusBraid it closes from.

TorusKnot(p, q)$T(p,q)$ as a knot — the closed form, with no braid involved

Typed signatures

Type	Package	Applies
(integer, integer) -> expression<TorusKnot>	braid	always

Domain: Braids and knots

Examples

Basic #

$T(3,2)$ is the trefoil

live ↗

In$\mathrm{AlexanderPolynomial}(\mathrm{TorusKnot}(3, 2))$Out$t^2-t+1$

live ↗

In$\mathrm{JonesPolynomial}(\mathrm{TorusKnot}(2, 5))$Out$-\frac{1}{t^7}-\frac{1}{t^5}+\frac{1}{t^6}+\frac{1}{t^4}+\frac{1}{t^2}$

About

Examples	2

See also: TorusBraid, AlexanderPolynomial, JonesPolynomial