Backlund transformations and knots of constant torsion

TL;DR

This paper explores how Backlund transformations for pseudospherical surfaces induce transformations on space curves with constant torsion, leading to new analytic representations of various knot types, especially in the context of vortex filament flow.

Contribution

It introduces a method to generate analytic constant-torsion representatives for numerous knot types using Backlund transformations, linking surface theory and knot geometry.

Findings

Derived analytic solutions for constant-torsion knots.

Connected Backlund transformations to vortex filament dynamics.

Provided new representations for complex knot types.

Abstract

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.