Knot Group Symmetries

This page is mostly an excuse to show off something cool I made during my summer working with Ben Williams that would not show up in a paper. At one point during the summer I was studying how the symmetries of knot 9-48 act on its corresponding fundamental group. Generally, the way you do this is quite geometric: just draw out the generating loops of the fundamental group and see where they end up under the symmetric transformation. The issue is this does not work with knot 9-48 as there are no symmetries of the knot that can be seen in \(\mathbb{R}^3\) alone. We instead need to work with \(\mathbb{R}^3\) plus an added point at infinity, and symmetries that look like this.

My solution to the problem was simple but time-consuming: create a 3d graphing program which plotted surfaces and curves projected into \(\mathbb{R}^3\) from \(\mathbb{S}^3\), and use the coordinates in \(\mathbb{S}^3\) to more easily apply the required transformations. The early tests of this were with a much simpler (5,2)-torus knot.

Afterwards I applied the program to knot 9-48 with an extra rotation about the z-axis.

Using this I was able to determine the action of the transformation on the corresponding knot group for knot 9-48. If you are interested in the gritty details (along with all the rest of the calculations I did that summer) see the attached pdf.

I will end this off with a couple of links: The finished product for knot 9-48 and a template for those of you crazy enough to try and do something like this yourself.