Adams operators and knot decorations

TL;DR

This paper connects Adams operators from quantum group representations to knot invariants via skein theory, explicitly identifying their images and analyzing their effects on Vassiliev invariants in knot theory.

Contribution

It explicitly determines the skein element corresponding to Adams operators on fundamental representations and links this to Vassiliev invariants of knots colored by these operators.

Findings

Identified the skein element as a linear combination of simple m-string braids.

Connected Adams operators to the Vassiliev invariants' weight systems.

Provided a formula for the Vassiliev invariant of knots colored by Adams operators.

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, \psi_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 \psi_m(c_1) is the canonical Vassiliev invariant with weight system W_n\psi_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 \psi_m^{(n)} is the Adams operator on n-chord diagrams defined by Bar-Natan.