Periodic discrete Darboux transforms

TL;DR

This paper introduces a quaternionic formalism for discrete Darboux transforms of polarised curves, enabling linearisation of monodromy and providing explicit parametrisations for special cases like circles and bicycle correspondences.

Contribution

It presents a novel quaternionic approach to discrete Darboux transforms, simplifying analysis and deriving explicit formulas for closed cases.

Findings

Linearisation of monodromy for discrete Darboux transforms

Explicit parametrisations for discrete circles and bicycle correspondences

Extension of the method to closed transforms and correspondences

Abstract

We express Darboux transformations of discrete polarised curves as parallel sections of discrete connections in the quaternionic formalism. This immediately leads to the linearisation of the monodromy of the transformation. We also consider the integrable reduction to the case of discrete bicycle correspondence. Applying our method to the case of discrete circles, we obtain closed-form discrete parametrisations of all (closed) Darboux transforms and (closed) bicycle correspondences.