Written by Sion Jang and Charlotte Peete (students in Math 383D Knot Theory Spring 2023).
The mosaic number for the 71 knot is six, meaning that it cannot be created on a grid using mosaic tiles smaller than 6 x 6. We are creating a 3D version of 71 mosaic knot using Cinema 4D as our main design program. We created this knot using the same method as our Trefoil knot. However, we made changes to the Chamfering process, the diameter of the tube, and the distance between some of the over-strands and feet.
When creating this knot with the same process as the Trefoil and Figure-8 knots, we came across a few problems.
Since the 71 knot has more crossings than either of the other two knots we created, we found that the close proximity of the feet would lead to self-intersections. We first changed the diameter of the circle from 6 to 4 mm. While this change helped to solve the intersection problem, the feet still seemed to be close together. Aesthetically, we still weren’t satisfied with how crowded the feet looked. So, we changed the z-coordinates of the horizontal feet to create more space between the adjacent vertical feet. The arrows in Figure 1 point to the three feet for which we changed these coordinates. Figure 2 shows an overhead view of the spacing between the feet with these changes.
The biggest challenge we came across with this knot was figuring out how to properly curve vertices without distorting the rest of the knot. Our original method of Chamfering did not work because there wasn’t enough space between the curves of the knot and the feet. To fix this problem, we added an additional point next to each vertex of the over-strand immediately before the foot. These points were added as close to the original vertices as possible.
Figure 3 shows our final product.