We find new solutions to the Yang-Baxter equation in terms of the interwiner matrix for semi-cyclic representations of the quantum group Uq(sl(2)) with q = e2πi/N. These interwiners serve to define the Boltzmann weights of a lattice model, which shares some similarities with the chiral Potts model. An alternative interpretation of these Boltzmann weights is as scattering matrices of solitonic structures whise kinematics is entirely governed by the quantum group. Finally, we consider the limit N → ∞ where we find an infinite-dimensional representation of the braid group, which may give rise to an invariant of knots and links.