Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy

We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie on the sphere S^3, ii) the distance between any two subsequant points does not vary in time, iii) the dynamics does not depend explicitly on the radius of the sphere. These results generalize to a discrete context our previous work on continuous curves.