New Torus Link, Improved Visualizations, and Cinema 4D Problems.

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

We created the T(2, 8) torus link (or 821 link) using Cinema 4D. The equations for this two component link are x = Cos[t]*(3+Cos[4t]), y = Sin[t]*(3+Cos[4t]), and z = Sin[4t]. The second component is created by the equations x = Cos[t]*(3+Cos[4t+Pi]), y = Sin[t]*(3+Cos[4t+Pi]), and z = Sin[4t+Pi].

We also created the T(3, 3) torus link (or 632 link). With the  632 link, each of the three components goes around the longitude once and goes around the meridian once. The equations for this knot are x1 = Cos[t]*(3+Cos[t]), y1 = Sin[t]*(3+Cos[t]), z1 = Sin[t], x2 = Cos[t]*(3+Cos[t+2*Pi/3]), y2 = Sin[t]*(3+Cos[t+2*Pi/3]), z2 = Sin[t+2*Pi/3], and x3 = Cos[t]*(3+Cos[t+4*Pi/3]), y3 = Sin[t]*(3+Cos[t+4*Pi/3]), z3 = Sin[t+4*Pi/3].

Image showing the T(3,3) torus link lying on the torus.

The T(3,3) torus link shown lying on the torus.

After creating the T(3, 3) or 632 link, we wanted to build a model that helps demonstrate what the torus link actually is. We did this by first opening back up the T(3, 6) or 632  in Cinema 4D. Then we created a torus surface and rotated it 90° so that the torus link sat in the right position on the torus surface. Then we changed the radius of both the meridian and the longitude so that the 3d model was in a presentable format. Our final model which is shown above gives a good physical representation of how a torus link is constructed.

We also created the  T(3, 6) torus link (or 633 link). With the 633 link, each of the three components goes once around the longitude and goes three times around the meridian. The equations for this knot are x1 = Cos[t]*(3+Cos[2t]), y1 = Sin[1t]*(3+Cos[2t]), z1 = Sin[2t], x2 = Cos[t]*(3+Cos[2t+2*Pi/3]), y2 = Sin[t]*(3+Cos[2t+Pi]), z2 = Sin[2t+2*Pi/3], and x3 = Cos[t]*(3+Cos[2t+4*Pi/3]), y3 = Sin[t]*(3+Cos[2t+4*Pi/3]), z3 = Sin[2t+4*Pi/3].

While using the Cinema 4D software, the biggest problem we had was fixing the join at the end of two strands. In Cinema 4D, the join will sometimes not look correct. In order to fix this, we will first decrease the period of the parametric equations in order to make the join fully noticeable. We decreased it from 2π (~6.28) to 6.275.

Image showing the two ends of the torus link with too many points highlighted

Too many points are highlighted near the ends of the torus link.

We then try to highlight all the points at the end of the knot, in order to use the “stitch and sew” function. We came across problems when we accidentally highlighted other points not at the end. This is shown in the figure to the right. This would happen more often when we increased the sample size to a number that was higher than necessary (>300). This is the case in the image below. By decreasing the sample size, it made it easier to highlight just the end points of the knot, as shown below. At this point we were able to successfully use the “stitch and sew” function.

Image showing points highlighted along the ends of the torus link.

This image shows that just the end points of the two ends of the torus link are highlighted. This allowed us to successfully use the Stitch and Sew function to join the ends together.

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].

Walking down memory lane

In Fall of 2014 I taught Math 341 Introduction to Topology. As part of the class I had the students design and then print a topological object. For most students, this ended up being the highlight of the course.

10482770_4584418265143_4975820982335381144_n-1We spent a week of class in the IQ center under the guidance of David Pfaff. He showed us how objects can be viewed in the stereo 3D lab and gave us a crash course in Cinema 4D. Students then let loose their imaginations and creativity. Many students chose to learn about knots and links, ribbon knots, and Seifert surfaces of knots and links. They produced some wonderful models. Other students chose to create objects with symmetry (like the 20 sided die), or the cube-like Cayley graph.

It turned out that getting the objects that could be 3d printed was hard work! Many objects had not been optimally made (for example with normal vectors pointing inwards). We were fortunate to have David Pfaff’s expertise in sorting out these errors. Eventually all the objects were printed using the IQ center’s ProJet 260. Some of them needed to be printed twice, as they broke when being removed from the printer. Many 3d printed math objects from this class and from Aaron Abrams first year seminar currently reside in the Mathematics Department.