Tree invariants and Milnor linking numbers with indeterminacy

TL;DR

This paper explores the relationship between tree invariants of string links and Milnor linking numbers, establishing their invariance under link homotopy and providing a recursive method for computation.

Contribution

It demonstrates that certain residue classes of tree invariants are link homotopy invariants and aligns their indeterminacy with Milnor's, offering a geometric perspective and computational approach.

Findings

Residue classes of tree invariants are link homotopy invariants.

Indeterminacy of tree invariants matches Milnor's indeterminacy.

Provides a recursive procedure for computing arrow polynomials.

Abstract

The paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak and closely related to the classical Milnor linking numbers also known as $\bar{\mu}$--invariants. We prove that, analogously as for $\bar{\mu}$--invariants, certain residue classes of tree invariants yield link homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor's indeterminacy. For practical purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.