Suppose that a flexible circular wire is twisted and folded back on itself to form (nearly) the double cover of a circle, then dipped in soap solution in such a way as to create a soap film in the form of a Möbius strip. Suppose now that the wire is slowly untwisted and unfolded back towards its original circular form. At a certain critical stage in this process, the film jumps from the one-sided Möbius strip to a two-sided surface spanning the wire. We have analysed both experimentally and theoretically how this topological jump occurs. This involves consideration of the role of the finite cross-section of the wire, no matter how small this may be. The surface before the jump may be idealised as the (minimum area) incomplete 'Meeks surface', which becomes unstable at a critical value of its defining parameter.

This topological jump is, in certain respects, analogous to the jump that occurs when a circular vortex (or magnetic flux) tube is twisted to the form of a figure-of-eight and forced to reconnect to form two separate tubes through viscous diffusion. This process will also be described, and it will be shown that helicity, a topological invariant of the ideal Euler equations, is no longer invariant during such a reconnection process.

References (downloadable here):
R. E. Goldstein, J. McTavish, H. K. Moffatt, and A. I. Pesci. Boundary singularities produced by the motion of soap films. Proc. Natl. Acad. Sci. 111 (23):8339-8344, 2014.
Y. Kimura and H. K. Moffatt. Reconnection of skewed vortices. J. Fluid Mech. 751:329-345, 2014.
H. K. Moffatt. Helicity and singular structures in fluid dynamics. Proc. Nat. Acad. Sci. 111 (10), 3663-3670, 2014.