Milnor invariants of string links, trivalent trees, and configuration space integrals

TL;DR

This paper links Milnor's homotopy link invariants with trivalent trees and configuration space integrals, providing a combinatorial framework and partial formulas for their explicit computation.

Contribution

It establishes a correspondence between Milnor invariants and trivalent trees using configuration space integrals, and develops a combinatorial analysis for explicit formulas.

Findings

Milnor invariants correspond to linear combinations of trivalent trees.

Configuration space integrals convert diagram products into invariant products.

Partial recipes for explicit integral formulas for Milnor invariants are provided.

Abstract

We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent "homotopy link diagrams" which corresponds to all finite type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas for products of Milnor invariants from trivalent forests. We also obtain cohomology classes in spaces of link maps from the same data.