The Backlund transformation for pseudospherical surfaces, which is equ
ivalent 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.