The Chiral Potts Model and Its Associated Link Invariant

TL;DR

This paper introduces a novel link invariant derived from the exactly solvable chiral Potts model, utilizing a generalized Gaussian summation identity to express the invariant in a tractable form, distinct from traditional quantum group invariants.

Contribution

It establishes a new link invariant based on the chiral Potts model and demonstrates its computation using a generalized Gaussian summation, expanding the class of known link invariants.

Findings

The invariant is expressed as a lattice sum associated with the link diagram.

The invariant involves roots of unity and differs from quantum group invariants.

A comprehensive table of invariants for links up to 8 crossings is provided.

Abstract

A new link invariant is derived using the exactly solvable chiral Potts model and a generalized Gaussian summation identity. Starting from a general formulation of link invariants using edge-interaction spin models, we establish the uniqueness of the invariant for self-dual models. We next apply the formulation to the self-dual chiral Potts model, and obtain a link invariant in the form of a lattice sum defined by a matrix associated with the link diagram. A generalized Gaussian summation identity is then used to carry out this lattice sum, enabling us to cast the invariant into a tractable form. The resulting expression for the link invariant is characterized by roots of unity and does not appear to belong to the usual quantum group family of invariants. A table of invariants for links with up to 8 crossings is given.