Braid group representations arising from the generalized chiral Potts models
A class of braid group representations are constructed for each nonsingular bilinear form \(B:(\mathbb{Z}/N\mathbb{Z})^ l\times(\mathbb{Z}/N\mathbb{Z})^ l\to\mathbb{Z}/N\mathbb{Z}\) with \(N\) odd. Let \(q\) be a primitive \(N\)-th root of unity, \(t\) a braid on \(M\) strings with \(v\) crossings and \(K=v-M+1\). Let further \(P=(p_{ij})_{1\leq i,j\leq K}\) denote a Seifert matrix for the closure of \(t\). Then associated link invariants \(\tau(t)\) are expressed as \[ \tau(t)=N^{-Kl/2}\sum_{a_ 1,\dots,a_ K}q^ Q,\quad Q=\sum_{1\leq i,j\leq K}p_{ij}B(a_ i,a_ j). \] For \(N\) an odd prime, they can be evaluated in terms of the classical Alexander polynomials if the quadratic form \(P\otimes B+{^ tP}\otimes{^ tB}\) is nonsingular. With a special choice of \(B\) these representations are Yang-Baxterized to the \({\mathfrak sl}(n)\) generalization of the chiral Potts model.
scientific article; zbMATH DE number 165418