# On the indeterminacy of Milnor's triple linking number

## TL;DR

This paper investigates the total triple linking number, an invariant that refines Milnor's classical triple linking number, demonstrating it is non-trivial for links with six or more components, thus capturing more link information.

## Contribution

It computes the total Milnor quotient and proves its non-triviality for links with at least six components, extending understanding of link invariants beyond classical triple linking numbers.

## Key findings

- Total triple linking number is non-trivial for links with ≥6 components.
- The total Milnor quotient contains more information than classical invariants.
- Classical triple linking numbers may be undefined while the total invariant is well-defined.

## Abstract

In the 1950's Milnor defined a family of higher order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received and fruitful study since its inception. In the case that $L$ has vanishing pairwise linking numbers, this triple linking number gives an integer valued invariant. When the linking numbers fail to vanish, this invariant is only well-defined modulo their greatest common divisor. In recent work Davis-Nagel-Orson-Powell produce a single invariant called the total triple linking number refining the triple linking number and taking values in an abelian group called the total Milnor quotient. They present examples for which this quotient is nontrivial even though none of the individual triple linking numbers are defined. As a consequence, the total triple linking number carries more information than do the classical triple linking numbers. The goal of the present paper is to compute this group and show that when $L$ is a link of at least six components it is non-trivial. Thus, this total triple linking number carries information for every $(n\ge 6)$-component link, even though the classical triple linking numbers often carry no information.