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The discrete Frenet equation entails a local framing of a discrete, piecewiselinear polygonal chain in terms of its bond and torsion angles. In particular,the tangent vector of a segment is akin the classical O(3) spin variable. Thusthere is a relation to the lattice Heisenberg model, that can be used to modelphysical properties of the chain. On the other hand, the Heisenberg model isclosely related to the discrete nonlinear Schr\"odinger (DNLS) equation. Herewe apply these interrelations to develop a perspective on discrete chainsdynamics: We employ the properties of a discrete chain in terms of a spinorialrepresentation of the discrete Frenet equation, to introduce a bi-hamiltonianstructure for the discrete nonlinear Schr\"odinger equation (DNLSE), which wethen use to produce integrable chain dynamics.

On relation between discrete Frenet frames and the bi-Hamiltonian structure of the discrete nonlinear Schr\"odinger equation