Beads: From Lie algebras to Lie groups

TL;DR

This paper explores the relationship between beads in the rational form of the Kontsevich integral and functions on Lie groups, providing a new rational form for the colored Jones function of knots.

Contribution

It introduces a rational form of the Kontsevich integral with beads and explains its connection to Lie group functions, advancing knot invariant theory.

Findings

Rational form of the Kontsevich integral with beads is constructed.

Relation between beads and Lie group functions is clarified.

A rational form for the colored Jones function is provided.

Abstract

The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function) with values in the symmetric algebra of the Lie algebra. Recently A. Kricker and the author constructed a rational form of the Kontsevich integral which takes values in an algebra of trivalent graphs with beads. After replacing beads by an exponential legs, this rational form recovers the Kontsevich integral. The goal of the paper is to explain the relation between beads and functions defined on a Lie group. As an application, we provide a rational form for the colored Jones function of a knot, conjectured by Rozansky. This is a revised version.