  The Backlund transformation for pseudospherical surfaces, which is equivalent
to that of the sine-Gordon equation, can be restricted to give a transformation
on space curves that preserves constant torsion. We study its effects on closed
curves (in particular, elastic rods) that generate multiphase solutions for the
vortex filament flow (also known as the Localized Induction Equation). In doing
so, we obtain analytic constant-torsion representatives for a large number of
knot types.
