Tag Archives: Torus Knot

Torus knots on tori

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A white donut shaped surface with a groove on the surface following a smooth curve

The T(3,2) torus knot shown on the torus.

My summer research student Kyle Patterson was interested in printing torus knots together with the tori that they lie on. Recall that torus knots are knots (and links) that can be moved so that they are embedded on a torus (or doughnut shape surface). Kyle constructed torus knots and links using the techniques described in this earlier blog post and this post too.

grey closed curved tube against a black background

T(3,2) torus knot with out the torus.

If we focus our attention on the T(3,2) torus knot, then this knot winds 3 times along the long way and 2 times along the short way around the torus. This knot has the following parametrization: \[ x(t)=\cos(3t)(3+\cos(2t)), \ y(t)= \sin(3t)(3+\cos(2t)), \ z(t) = \sin(2t)\] for t in [0,2π]. We note that the torus this knot lies on is defined by two circles.  We take a large/ring circle of radius 3 (given by equations x2+y2=9, z=0). We make each point of the ring circle the center of a small/pipe circle of radius 1 so that the plane of the pipe circle is perpendicular to the plane of the ring circle.  To construct this torus in Cinema 4D, simply add the Torus shape, then go to the Attributes menu and set the Ring Radius to 3cm and the Pipe Radius to 1cm. In order to have a smooth looking torus we need to change the number of Ring Segments to 150 and the number of Pipe Segments to 50.

grey donut shape with a groove cut into it along a curve against a black background

The torus minus the T(3,2) torus knot.

To finish the construction we used the Boole tool in two different ways. First, we took the union of the knot and the torus to create a new surface which reveals the knot lying on the torus. Second, we took the difference of the torus and the knot. This created a torus surface with a smooth groove in the surface following the path of the knot.

We now have three different kinds of files that can be printed. We could print the knot, the torus with the knot, and a torus with a smooth groove following the knot.  We printed the torus with the smooth groove following the T(3,2) torus knot. We also created a 3D print where the knot is one color and the torus is a different color. To do this we use the torus with groove file and the knot file and the Ultimaker5 3D printer in the IQ center at WLU.

We repeated this construction for the T(2,3), T(3,3) and T(3,6) torus knots and links.

White doughnut shape with a red smooth curve on the surface against a brown background

A red T(3,2) torus knot on a white torus.

 

Hyperboloidal Representation of Torus Knots

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Written by Timi Patterson (2024 Summer Research Scholar), added to by Elizabeth Denne.

A polygonall (3,5) torus knot arranged on a hyperboloid of one sheet

A T(3,5) torus knot arranged on two hyperboloids of one sheet.

In his book Knots and Links, Peter Cromwell details a representation of torus knots embedded in a parameterization on the union of two hyperboloids. He provides these instructions in Section 1.5:

Choose an angle θ in [0, π/2] and construct two points: \[ A=(\cos\theta, -\sin\theta, -1), \ \ B=(\cos\theta, \sin\theta, 1).\] The straight line through A and B is defined by \[x=\cos\theta, y=z\sin\theta.\] Rotating this line about the z-axis gives the hyperboloid \[x^2+y^2-z^2\sin^2\theta = \cos^2\theta.\] Let Ht denote the annulus obtained by restricting z between the interval from -1 to 1. The boundary curves of the annulus are unit circles: \[ x^2+y^2=1, \ \ z=\pm 1.\] Take the union of two of these Ht annuli with different values of t=theta. This new surface is topologically a torus.

The (p,q) torus knot with p strictly greater than q (and q greater than or equal 2) can be embedded in one of these “hyperboloidal” tori as follows. Choose t=theta and s=phi such that \[ \frac{q}{p}\cdot \frac{\pi}{2} <\theta < \min\{ \frac{\pi}{2}, \frac{q}{p} \pi\} \ \ \ \text{and} \ \ \ \phi = \frac{q}{p}\pi -\theta  .\]  The knot will lie in the torus which is the union of Ht and Hs.

Now take i in {0, 1, 2, … , 2p}. If i is odd, the vertices of the knot are \[ v_i=( \cos((i-1)\pi\frac{q}{p} + 2\theta), \sin((i-1)\pi\frac{q}{p}+2\theta), 1), \] and if i is even, the vertices of the knot are \[v_i =  (\cos((i\pi\frac{q}{p}), \sin(i\pi\frac{q}{p}), -1) .\]

Black and white image of a knot made of 6 edges.

Polygonal (3,2) torus knot whose edges lie on hyperboloids of one sheet.

Following these instructions for the trefoil knot viewed as a T(3,2) torus knot, with\[ \theta = \frac{2\pi}{5} \ \ \text{and} \ \  \phi = \frac{4\pi}{15},\]

I constructed the following vertices:
x y z
1 -0.8090169944 0.5877852523 1.0
2 -0.5 -0.8660254038 -1.0
3 0.9135454576 0.4067366431 1.0
4 -0.5 0.8660254038 -1.0
5 -0.1045284633 -0.9945218954 1.0
6 1.0 0.0 -1.0
Black and white photo of a 4 stick unknot

One component of the T(4,2) torus link.

I first constructed the T(5,3) torus knot in Cinema 4D, as the vertices were detailed in the book. I did this by creating splines using the vertices created by the functions, and using Cinema 4D’s sweep function to create a model with a thickness. I used the Chamfer tool to smooth out the corners. I then went on to create the trefoil knot, the T(10,8) torus knot, the T(4,2), T(12,3), and T(10,8) torus links all in Cinema 4D with the same technique.

Black and white image of 8 line segments, some connected.

The T(4,2) torus link. Note the two components differ by a 90 degree rotation.

To create the links, I had to separate the functions into the different components. Take for example the T(4,2) torus link. When evaluating the formulas, the q/p reduces to 1/2. To then create the two different components of T(2,1), the first component uses the vertices constructed as described above. To construct the vertices of the second component, simply add π/2 to the inside of the trig function in each component. (For example cos(iπ/2 +π/2) for the first term in the even index vertex.) Therefore, I had two components with these vertices (for theta=3π/8 and phi=π/8).

Component 1:

x y z
1 -0.7071067812 0.7071067812 1.0
2 -1.0 0.0 -1.0
3 0.7071067812 -0.7071067812 1.0
4 1.0 0.0 -1.0

Component 2:

x y z
1 0.7071067812 0.7071067812 1.0
2 0.0 -1.0 -1.0
3 -0.7071067812 -0.7071067812 1.0
4 0.0 1.0 -1.0
black and white image of a large number of intersecting line segments

One version of the T(10,8) torus link.

A black and white image of a series of nested interleaved line segments.

A different version of the T(10,8) torus link.

One thing that I experimented with some when working with the T(10,8) torus link is manipulating the theta value to try and reduce any intersections of the model. I created two different models, one with theta=5π/11, the other with theta=5π/12. They varied a lot with where the self-intersections of the tubes were, but alas both of the tubes did self intersect. That will most likely happen with a lot of torus knots or links with p’s and q’s of closer value, but some self-intersections may be able to be avoided by manipulating the theta value,

White knot on brown background

The T(10,3) torus knot.

White link on brown background

The T(10,8) torus link.

I then went on to print out most of these 3D models. The T(10,3) torus knot and T(10,8) torus link are shown above. It turned turns out that these models are surprisingly difficult to print. Take a look at the models. There is only a small area on the base of each V shape. The edges of the knots have to “grow” from this small base. This means the models are unstable. So even though the angle of the edges is high with respect to the ground, the models still need support. We printed several without supports and had some spectacular failures, as shown below. After the edges of the knots fell over on the print bed, the printer kept going leaving a squiggly mess of filament.  The solution to this problem was to increase the angle for the supports from 43 to 55 degrees.

two distinct white spiky shapes on black platforms

Several of the builds failed as the edges of the knots fell over during 3D printing.

 

 

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

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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.

Knots as ribbons

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I’ve continued with my project to edit the 3D printed models my Fall 2014 Math 341 Introduction to Topology class made. Recently, I came across two of my favorite pieces. The first is a model that was designed by Emily Jaekle (’16) and is a ribbon version of the (3,5) torus knot.

IMG_4407    IMG_4458

This \((5,3)\) ribbon torus knot was designed entirely in Cinema4D. The curve was created using the Formula tool with parametrization \(x(t)=(2+\cos(5t))\cos(3t), y(t)=(2+\cos(5t))\sin(3t), z(t)=-\sin(5t)\) for \(t\in[-\pi, \pi]\). The trianglulated surface was created by first adding in a small rectangle, then using the SweepNurbs (without caps). The rectangle was also rotated 1800 degrees in the process. The small gap was fixed using the Bridge tool in Edge mode. This ribbon knot was originally printed in blue on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in black resin. You can find the model here on Thingiverse.

The second model was designed by Cathy Wang (’15) and is a ribbon version of the (3,2) trefoil knot with an amazing color scheme.

IMG_4404The entire model was designed in Cinema4D. The knot was created using the Formula tool with parametrization \(x(t)=(2+\cos(2t))\cos(3t), y(t)=(2+\cos(2t))\sin(3t), z(t)=-\sin(2t)\) for \(t\in[-\pi,\pi]\). The trianglulated surface was created by first adding in a small rectangle, then using the SweepNurbs (without caps). The width and height of the rectangle was adjusted so the band is not a constant size through the knot. The overlapping edges and small gaps were also fixed. Finally, the knot was colored with a beautiful rainbow-gradient. This ribbon knot was originally printed in rainbow colors on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in black resin. You can find the model here on Thingiverse.

Math at the Simon’s Center for Geometry and Physics

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IMG_3442I had the very great privilege of being a co-organizer of a workshop held at the Simon’s Center for Geometry and Physics and NYU Stony Brook. This was the workshop on the Symplectic and Algebraic Geometry in the Statistical Physics of Polymers. It was my first time to this campus, and I had a blast with both the math at the workshop AND all the visualization of math in the environment.

My first hint that things were going to be special, was the fantastic Umbilic Torus sculpture found at the end of an avenue of trees between the center and the math department.

IMG_3434The sculpture is by Dr Helamun Ferguson, click  here to find a photo gallery showing the design and construction of the piece. 

The sculpture consists of a space filling curve all over the surface of the sculpture. The sharp curve along the edges is a trefoil knot, winding three times around the central hole (the longitude on the torus) and twice around the sculpture the other way (the meridian on the torus).

The base of the sculpture is a large round granite disk with a 3 sided deltoid mirroring the 3-fold symmetry of the sculpture overhead. The base had to be left to settle for a year, and was greatly loved by the local skate-boarders!

84-umbilictorus-5392IMG_3435

The Simon’s Center itself is in a wonderful airy building, with mathematical themes blended seamlessly in the design. I kept finding treasures as the workshop went on. The most obvious, is the sandstone wall behind the stair case leading up to the cafe on the second floor. It is covered with small math motifs from knots, to physics, to finding the square root of 2.

IMG_3443IMG_3453

Even the screens on the side of the first floor lounge are mathematical, with different tilings of the plane illustrated. Just love the artistry of the designs in them.

IMG_3465 IMG_3464 IMG_3462

 

Torus Knots

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I modeled the trefoil knot as two torus knots \(T(2,3)\) and \(T(3,2)\). The parametric equations for a \(T(p,q) \) knot are \(x = \cos(pt)*(3+\cos(qt)), y=\sin(pt)*(3+\cos(qt)) \), and \(z=\sin(qt) \). Here, \(p\) is the number of times the knot winds around the longitude of a torus, and \(q\) is the number of times the knot winds around the meridian of a torus.

2-3-torus-2Both models were printed on the FormLabs printer. I first made a small \(T(2,3) \) knot with a label extruded out of the curve (as shown to the left). I used Cinema 4D to design the model by using the Formula Spline to draw the curve, the Sweep NURB to give the curve depth, and the Wrap Tool to wrap the text around the curve. I also used the Extrude Tool to give the equations depth and the Boole Tool to connect the equations to the curve. For both knots I had to make sure the ends of the knots overlapped correctly. Before printing the \(T(3,2)\) knot, I had to change the range of \(t\) to \(t=[0, 2\pi] \) instead of \(t=[0, 5\pi]\) (I initially used \(5\pi\) to be sure that the curve closed).

3-2-torus-2The first \(T(2,3)\) knot came out nicely, however the text was a little small. Using the subscript made the numbers too small, so I reprinted the knot used parentheses instead, as shown here. The \(T(3,2)\) knot also looked great, as it was smooth and there were merely small nubs where the supports were, which could be removed with an exacto blade. We’ve discovered that the FormLabs printer makes smoother surfaces and finer curves than does the MakerBot, which is why it is ideal for printing knots.

You can find the torus knots on Thingiverse here T(2,3) and here T(3,2). Instructions on how to make torus knots in Cinema 4D can be found here. Professor Denne has also created another worksheet in Mathematica about Torus knots. It can be found here.