PERIODIC RESULTS FOR THE COLOURED (GENERALISED) JONES POLYNOMIAL

Abstract

Using the vertex model interpretation of the coloured (generalised) Jones polynomial of a link L, we show that if the colour of the ith component is Ni+mir, then modulo tr−1 this coloured Jones polynomial is congruent, up to a product of calculable factors, to the coloured Jones polynomial with the colour of the ith component Ni, where Ni and r are positive integers, and mi is a non-negative integer. The proof depends on the fact that, up to a known factor, the coloured Jones polynomial of a link may be calculated from a (1, 1)-tangle, the closure of which represents the link.