Abstract 

The Temperley-Lieb algebra has far reaching applications in both mathematics and statistical mechanics, where it underpins a number of key models, notably in the spin-$\frac{1}{2}$ XXZ and $N$-state Potts quantum spin chains. We introduce a new coupled Temperley-Lieb algebra consisting of $N$ types of Temperley-Lieb generators. The $\mathbb{Z}_{N}$ clock Hamiltonian including the $N$-state superintegrable chiral Potts Model is written in terms of $N-1$ types of generators of the algebra. We present a diagrammatic representation of the coupled Temperley-Lieb algebra in terms of a planar parafermion algebra. A planar string Fourier transform is defined and applied in the derivation of cubic relations in the coupled algebra. The superintegrable Hamiltonian in the planar representation provides a diagrammatic proof of the Dolan-Grady relations and the resultant Onsager algebra as well as the existence of parafermion shift modes in the $\mathbb{Z}_{N}$ clock models in general. Baxterization of the coupled algebra in the limiting cases of the Fateev-Zamolodchikov and Baxter's $\mathbb{Z}_{N}$ parafermion chain is also considered.