Abstract or Additional Information
The M\"obius energy is an object of interest at the intersection of physical knot theory and geometric/harmonic analysis. In their seminal paper, Freedman-He-Wang proved that measuring the energy of a given simple closed curve can yield information regarding the knot's classical topological invariants. Furthermore, they also proved that minimizing knots are $C^{1,1}$. That is, $C^1$ with Lipschitz derivative. A series of increasingly strong regularity theorems were proven, culminating in the remarkable result by Blatt-Reiter-Schikorra and others that critical points are analytic, the first analyticity result of a nonlocal operator to our knowledge. A heuristic observation of the Gateaux derivative led me to investigate the M\"obius energy of helices, and the analyticity result indicated that complex analysis could prove to be useful. Calculating the asymptotics of the helix as it coils and uncoils reveals a nice parable about the duality of pure and applied mathematics