Adderton, RemyBatchelor, MurrayWedrich, Paul2022-10-072022-10-071751-8121http://hdl.handle.net/1885/274374The Hamiltonian of the N-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by N − 1 types of Temperley-Lieb generators. This generalises a previous result for N = 3 obtained by Fjelstad and Mansson (2012 J. Phys. A: Math. Theor.45 155208). A pictorial representation of a related coupled algebra is given for the N = 3 case which involves a generalisation of the pictorial presentation of the Temperley-Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the N = 3 SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight √3 and weight 2, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter ρ = e2πi/3 for the SICP chain and ρ = 1 for the staggered XX chain. These ρ values are derived assuming the Kauffman bracket skein relation.This work has been supported by the Australian Research Council Discovery Project DP180101040 and by the National Natural Science Foundation of China Grant No. 11575037.PW was supportedby the National Science Foundationunder Grant No. DMS-1440140, while inresidence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester.application/pdfen-AU© 2020 The Author(s).https://creativecommons.org/licenses/by/4.0/2020-08-1310.1088/1751-8121/aba1432021-11-28Creative Commons Attribution 4.0 licence