The Kontsevich integral and algebraic structures on the space of diagrams

TL;DR

This paper explores the algebraic structures of the diagram space in the Kontsevich integral for knots, providing explicit calculations for torus knots and linking these to key theorems in knot theory.

Contribution

It describes the algebraic structures on the diagram space and applies them to derive explicit formulas for the Kontsevich integral of torus knots up to degree five.

Findings

Explicit expression for the Kontsevich integral of torus knots up to degree five

General coefficients of wheel diagrams in the Kontsevich integral

Application of Le's theorem and Melvin-Morton Theorem to knot invariants

Abstract

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le's theorem on the behaviour of the Kontsevich integral under cabling and with the Melvin-Morton Theorem, to obtain, in the Kontsevich integral for torus knots, both an explicit expression up to degree five and the general coefficients of the wheel diagrams.