Geometric and topological aspects of vortex filament dynamics under LIA

Geometric and topological aspects associated with integrability of vortex filament motion in the Localized Induction Approximation (LIA) context (which includes a family of local dynamical laws) are discussed. We show how to interpret integrability in relation to the Biot-Savart law and how soliton invariants can be interpreted in terms of global geometric functionals of knotted solutions. Under the basic (zeroth-order) LIA, we prove that vortex filaments in the shape of torus knots T p, q (p, q co-prime) with (q/p)>1 are stable, whereas those with (q/p)<1 are unstable.