  In this paper, we consider the evolution of the Vortex Filament equation
(VFE): \begin{equation*} \mathbf X_t = \mathbf Xs \wedge \mathbf Xss,
\end{equation*} taking $M$-sided regular polygons with nonzero torsion as
initial data. Using algebraic techniques, backed by numerical simulations, we
show that the solutions are polygons at rational times, as in the zero-torsion
case. However, unlike in that case, the evolution is not periodic in time;
moreover, the multifractal trajectory of the point $\mathbf X(0,t)$ is not
planar, and appears to be a helix for large times.
  These new solutions of VFE can be used to illustrate numerically that the
smooth solutions of VFE given by helices and straight lines share the same
instability as the one already established for circles. This is accomplished by
showing the existence of variants of the so-called Riemann's non-differentiable
function that are as close to smooth curves as desired, when measured in the
right topology. This topology is motivated by some recent results on the
well-posedness of VFE, which prove that the selfsimilar solutions of VFE have
finite renormalized energy.
