The Helicity of a Knotted Vortex Filament

摘要: The helicity H associated with a knotted vortex filament is considered. first constructed starting from circular tube, in three stages involving injection of (integer) twist, deformation and switching crossings. This produces tube the form an arbitrary knot K; each line (trivial) satellite K, linking number any pair lines same integer n. It shown that these circumstances given by = nк 2 where к circulation tube. result discussed relation to earlier works, particular work Călugăreanu (1959, 1961) which establishes that, for twisted ribbon axis C n sum ingredients: $$\frac{H}{{{\kappa ^2}}} W(C) + T(C) \frac{1}{{2\pi }}{[\Delta \Theta ]_C}$$ where writhing total torsion. quantity [∆Θ] represents net angle rotation spanwise vector on relative Frenet triad one passage round C. Both are discontinuous deformations take through inflexion point. generic behaviour such point analysed clarified §6.