Milnor invariants and edge-homotopy classification of clover links

TL;DR

This paper introduces a new way to classify clover links using Milnor invariants, establishing conditions under which these invariants are well-defined and complete for edge-homotopy classification, especially for 3-clover links.

Contribution

It defines Milnor numbers for clover links via bottom tangles and proves their effectiveness in classifying clover links up to edge-homotopy and $C_{2k+1}$-equivalence.

Findings

Milnor numbers of length k or less vanish imply well-defined Milnor numbers of length 2k+1 or less.

Two clover links are edge-homotopic iff their Milnor numbers of length 2k+1 or less are equal.

Complete classification of 3-clover links using Milnor numbers of length 3 or less.

Abstract

Given a clover link, we construct a bottom tangle by using a disk/band surface of the clover link. Since the Milnor number is already defined for a bottom tangle, we define the Milnor number for the clover link to be the Milnor number for the bottom tangle and show that for a clover link, if Milnor numbers of length k or less vanish, then Milnor numbers of length 2k+1 or less are well-defined. Moreover we prove that two clover links whose Milnor numbers of length k or less vanish are equivalent up to edge-homotopy and $C_{2k+1}$-equivalence if and only if those Milnor numbers of length 2k+1 or less are equal. In particular, we give an edge-homotopy classification of 3-clover links by their Milnor numbers of length 3 or less.