Written by Sion Jang and Charlotte Peete (students in Math 383D Knot Theory Spring 2023).
Mosaic knot theory uses a combination of the following eleven tiles to create a knot or a link representation on an nxn grid. These tiles are shown in Figure 1. As explained in the Knot Mosaic Tabulation paper by Hwa Jeong Lee, Lewis D. Ludwig, Joseph S. Paat, and Amanda Peiffer, the mosaic number of a knot K is the smallest integer n for which K can be represented on an mosaic board. This mosaic number is a knot invariant and can be used to distinguish between two knots.
Figure 1: tiles used to create a mosaic knot or link.
The Knot Mosaic Tabulation Paper provided the minimal grid mosaic diagrams for all 36 prime knots of eight crossings or fewer. The mosaic number for a trefoil is four, m(31)=4. Thus, a trefoil cannot be created on a 3 × 3 grid, and the minimum grid needed is a 4×4 grid. We created a 3D version of 31 Trefoil mosaic knot using Cinema 4D as our main design program. To create this knot, we first drew a coordinate plane onto the diagram of the Trefoil knot, as shown in Figure 2.
Figure 2: Trefoil mosaic knot with coordinates in xy-plane.
We did this so that we could insert integer coordinates into Cinema 4D that would allow for accurate spacing in our knot. Our final knot was scaled to be around 7×7 cm without the circle sweep, which is the thick tube around the curve.
One of the main challenges we encountered in constructing this knot was figuring out how to represent the over and under-crossings. Using this blog by Laura Taalman as inspiration, we decided to create “feet” for our knot. The distance from the center of each foot to the center of the over-strand is 1 cm. We also put a .125 mm space on either side of the over-strand between the legs to show that the strands are not connected. Figure 3 shows a close-up representation of one of the feet and legs on this knot.
Figure 3: The foot and two legs create a strand which crosses under a different strand of the knot.
The coordinates we chose made the knot have an angular rather than curved shape, and we manually curved the knot in Cinema 4D using the “Chamfer tool” once our knot was constructed. Figure 4 shows our knot before we curved the edges. Figure 5 shows our knot after we curved the edges. For the different curves, we used different values of the Chamfer to get the desired look.
Figure 4: Top view & angled view of the trefoil knot before curving the corners.
Figure 5: Top view & angled view of the trefoil knot after curving the corners.
FIgure 6: The figure-8 mosaic knot.
Using this same process, we created 41, the Figure-8 knot. The final product is shown in Figure 6.