Torus Paths

An

(n,k)

torus path is a curve on the surface of a torus that winds

n

times around the hole and

k

times through the hole. In this Demonstration you can vary the radius

R

(thickness) of the torus and the values of

n

and

k

. To help understand the numbers

n

and

k

, you can show the extra elements "around", being a circle around the hole, and "through", being a circle through the hole. The number

n

is the number of times the curve intersects the "through" circle, and the number

k

is the number of times the curve intersects the "around" circle. The torus path can be thickened to a tube with radius

r

.

You can also show

p

equally spaced points on the torus path, and shift the points along the path. These points can be connected by lines and tubes.

A (2, 3) torus path forms a trefoil knot, and a (2, 5) torus path forms a surgeon's knot.