page_title: Chiral Potts model
section_title: Knot theory
hierarchical_section_title: Chiral Potts model / Relation with mathematics / Knot theory
caption_attribution_description: English: Property of the Weights
context_page_description: The chiral Potts model is a spin model on a planar lattice in statistical mechanics. As with the Potts model, each spin can take n=0, ...N-1 values. To each pair of nearest neighbor of spins n and n', a Boltzmann weight W is assigned. The model is chiral, meaning W≠ W. When its weights satisfy the Yang–Baxter equation, it is integrable. For the integrable chiral Potts model, its weights are parametrized by a high genus curve, the chiral Potts curve.
  Unlike the other solvable models, whose weights are parametrized by curves of genus less or equal to one, so that they can be expressed in term of trigonometric, or rational function or by theta functions, this model involves high genus theta functions, which are not yet well developed. Therefore, it was thought that no progress could be made for such a difficult problem. Yet, many breakthroughs have been made since the 1990s. It must be stressed again that the chiral Potts model was not invented because it was integrable but the integrable case was found, after it was introduced to explain experimental data. In a very profound way physics is here far ahead of mathematics. The history and its development will be presented here briefly.
context_section_description: The integrable chiral Potts weights are given in product form  as
  where ωᴺ=1 and we associate with each of the rapidity variable p with three variables (xₚ,yₚ,μₚ) satisfying
  It is easy to see that
  which is similar to Reidemeister move I. It was also known that the weights satisfying the inversion relation,
  This is equivalent to Reidemeister move II. The star-triangle relation
  is equivalent to Reidemeister move III. These are shown in the figure shown here.
caption: Property of the Weights: Reidemeister Move I
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