Adams operators and knot decorations

Abstract

We use an explicit isomorphism from the representation ring of the quantum group U_q(sl(N)) to the Homfly skein of the annulus, to determine an element of the skein which is the image of the mth Adams operator, ψ_m, on the fundamental representation, c_1. This element is a linear combination of m very simple m-string braids. Using this skein element, we show that the Vassiliev invariant of degree n in the power series expansion of the U_q(sl(N)) quantum invariant of a knot coloured by ψ_m(c_1) is the canonical Vassiliev invariant with weight system W_nψ_m^{(n)} where W_n is the weight system for the Vassiliev invariant of degree n in the expansion of the quantum invariant of the knot coloured by c_1 and ψ_m^{(n)} is the Adams operator on n-chord diagrams defined by Bar-Natan.