Linking coefficients and the Kontsevich integral

TL;DR

This paper provides a combinatorial formula linking linking coefficients and the Kontsevich integral, enabling the expression of products of invariants and sums of coefficients in terms of Kontsevich integral coefficients.

Contribution

It introduces a general combinatorial formula connecting linking coefficients and the Kontsevich integral, extending the understanding of how invariants are represented.

Findings

Expresses products of linking and framing invariants as Kontsevich integral coefficients

Relates sums of coefficients of a given degree to linking coefficients

Provides purely combinatorial proofs for these formulas

Abstract

It is well known how the linking number and framing can be extracted from the degree 1 part of the (framed) Kontsevich integral. This note gives a general formula expressing any product of powers of these two invariants as combination of coefficients in the Kontsevich integral. This allows in particular to express the sum of all coefficients of a given degree in terms of the linking coefficients. The proofs are purely combinatorial.