Conway polynomial and Magnus expansion

TL;DR

This paper explores the relationship between the Conway polynomial, Magnus expansion, and finite type invariants of pure braids, providing explicit mappings and conjectural generating functions involving multiple zeta values.

Contribution

It explicitly describes the mapping from chord diagrams on 3 strands to polynomials and evaluates it on the Drinfeld associator, linking braid invariants to multiple zeta values.

Findings

Explicit mapping from chord diagrams to polynomials

Evaluation on Drinfeld associator yields conjectural generating functions

Coefficients involve integer combinations of multiple zeta values

Abstract

The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We describe explicitly the resulting mapping from horizontal chord diagrams on 3 strands to univariate polynomials and evaluate it on the Drinfeld associator obtaining, conjecturally, a beautiful generating function whose coefficients are integer combinations of multiple zeta values.