Overview of Torus Shapes, Knots, and Links

Written by Hillis Burns, Shannon Timoney, Hall Pritchard (students in Math 383D Knot Theory Spring 2023).

Image of a torus with a 3 component link on it

The T(3,6) torus link.

The torus is the surface of a donut in 3-dimensions. A torus knot/link is a knot/link that can be moved to lay on the torus surface in R3.The image on the right shows a link being wrapped to lie on a torus; this is the T(3, 6) or 633 torus link. Knots are also commonly described in knot tables using the notation, Crnj. The crossing number is denoted by Cr, the number of components by n, and the certain configuration is j. As seen in this image, the torus has two key circles: the longitude, which wraps around the long way of the torus, and the meridian, which wraps around the short way. These are illustrated in the image below.

Image of a torus showing the longitude and meridian curves

A torus with some Longitude and Meridian curves highlighted.

The notation for torus knots is T(p, q); The knot wraps around the longitude p times, while it wraps q times around the meridian. The two figures are from a Mathematica file which visualizes torus links, and this website. This Knot Plot website also has a neat table showing many of the torus knots and links.

Using a Mathematica file provided by Professor Denne, we were able to start creating the T(2, 4) torus link (or 421) in Cinema 4D. This is a two-component link where each component goes once around the longitude and twice around the meridian, as illustrated below.

Image showing the T(2,4) torus link

The T(2,4) torus link.

The parametric equations that create the link are x =  Cos[t]*(2+Cos[2t]), y = Sin[t]*(2+Cos[2t]), z = Sin[2t], with t going from 0 to 2Pi. However, for this link there are two components. Thus, a second equation was needed for the second component. To create a torus link like this, the second equation must be rotated 180 degrees to fit with the first curve. To do that, we added Pi to the trigonometric equations of the first sweep: x = Cos[t]*(3+Cos[2t+Pi]),  y = Sin[t]*(3+Cos[2t+Pi]), z = Sin[2t+Pi].

We also decided to create the T(2, 11) torus knot (also known as 111) in addition to the links in Cinema 4D. This is a knot (one component link) where the curve goes twice around the longitude and 11 times around the meridian. The topmost figure below shows the original image of the torus knot that we created. The knot does not look smooth, as Cinema 4D only evaluated a few points along the parametrized curve. However, after adding more sample points we were able to make the torus knot smoother. The progression of sample points, from 20, to 50, to 100, to 200, is shown from top to bottom:

T(2,11) with 20 sample pointsT(2,11) with 50 sample points

T(2,11) with 100 sample pointsT(2,11) with 200 sample pointsWe also had to adjust the radius because the loops were too close together. In order to spread out the components, the radius was changed from 2 to 3.

Next, we created a T(2, 6) torus link (or 621) in Cinema 4D. With the 621, each of the two components goes once around the longitude and three times around the meridian. Using the Mathematica file, we knew that the equations for this link were, x = Cos[t]*(3+Cos[3t]),  y = Sin[t]*(3+Cos[3t]),  z = Sin[3t].  The second component’s equations, again rotated by Pi, consist of, x = Cos[t]*(3+Cos[3t+Pi]), y = Sin[t]*(3+Cos[3t+Pi]),  z = Sin[3t+Pi].