VORTEX FILAMENT IN A TREE-MANIFOLD AND ITS CLASSICAL PARTITION FUNCTION

The kinematics of a very thin vortex tube in a three-dimensional fluid may be described by the filament equation in the local induction approximation. $[1, 2] $ It is formulated as $\frac{\partial\gamma}{\partial t}=\frac{\partial\gamma}{\partial s}\cross\frac{\partial^{2}\gamma}{\partial s^{2}} $ , (1) where $\gamma=\gamma(t, s) $ denotes the position of the vortex filament in $\mathrm{R}^{3} $ with $t $ and $s $ being th