VIGRE Summer REU 2011
In the summer of 2011, I led a group of students in the University of Iowa's VIGRE-REU studying the Symmetries of Knots using techniques from abstract algebra and harmonic analysis. In this REU the students identified knots that are symmetric with respect to group actions, by constructing polygonal knot representations that are preserved by the group actions. They proved that the Fourier and Chebyshev approximations of the polygonal knots are smooth knots, and that these approximations preserve the desired symmetries.
Here is an explanation of what they did using pictures generated by their Maple code:
This is a figure 8 knot which has reflection-rotation axis symmetry, which means that it is preserved by rotating it 90 degrees around an axis and then reflecting in a plane perpendicular that axis. This symmetry is often encountered by chemists; molecules such as ethane and silicon tetrafluoride have this symmetry. Note that rotating this knot 90 degrees does not preserve the knot.
Even when they could not figure out the equations parameterizing a symmetric knot by hand, the students used the following algorithm for obtaining symmetric polygonal knots. They started with a knot that appeared to be nearly symmetric, and let a group, such as the dihedral group D_3, act on the knot. The action mapped a cloud of points near each vertex of the knot. This cloud of points is left invariant under the group action, and thus they took the average over each cloud to obtain a perfectly symmetric knot!
You can see the cloud of points around each vertex of this trefoil knot.
To obtain smooth knots with symmetries the students used Fourier and Chebyshev approximations of the symmetric polygonal knots. This is a picture of a Fourier approximation of a (11,1) torus knot.
Here is a picture of a Chebyshev approximation of the trefoil knot. Since Chebyshev approximations are a polynomial approximation, they give a "long knot," meaning that the strands of the knot only meet at the point at infinity. This means that they form a simple closed curve when embedded in the sphere but not in the plane.
A Chebyshev approximation of the (5,1) torus knot.