# Understanding the Evolution of the Vortex Filament Equation for Initial Polygonal Data

We consider the geometric flow

\[\mathbf X_t = \kappa\mathbf b,\]

where $\kappa$ is the curvature and $\mathbf b$ is the binormal component of the Frenet-Serret formulas. It can be expressed as

\[\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},\]

where $\wedge$ is the usual cross product, and $s$ denotes the arc-length parameter. This equation is known as the vortex filament equation (VFE).

Since the tangent vector $\mathbf T = \mathbf X_s$ remains with constant length, $\mathbf T$ can be assumed to take values on the unit sphere. Differentiating VFE, we get the Schroedinger equation on the sphere:

\[\mathbf T_t = \mathbf T \wedge \mathbf T_{ss}.\]

We consider the evolution of $\mathbf X(s, t)$ and $\mathbf T(s, t)$ for different types of initial data. On the one hand, this is well understood for planar regular polygons of $M$ sides and total length $2\pi$: Assuming uniqueness and bearing in mind the invariances and symmetries of the problem, we are able to fully characterize by algebraic means $\mathbf X(s, t)$ and $\mathbf T(s, t)$, at rational multiples of $t = 2\pi/M^2$. On the other hand, the situation becomes more involved when we consider nonregular polygons or polygons with nonzero torsion, and further research is needed here.