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The simplest model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation closely related to the cubic focussing nonlinear Schr(o)dinger equation.Closed finite-gap solutions of the vortex filament flow provide examples of evolv ing curves whose topological features can be related to their algebro-geometric description.In the second of two talks, we describe how the theory of isoperiodic deformations (developed by Grinevich and Schmidt, after Krichever) can be used to generate a family of closed finite-gap solutions of increasingly higher genus close to a multiply covered circle.Each step of the deformation process corresponds to constructing a cable on the previous filament, whose knot type is determined from the deformation scheme, and is invariant under the time evolution.