On the indeterminacy of Milnor’s triple linking number

In the 1950s 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 fruitful study since its inception. In the case that a link $L$ has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking number 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 each of the individual triple linking numbers take values in the trivial group, $\mathbb{Z}/1$, and so carry no information. As a consequence, the total triple linking number carries more information than do the classical triple linking numbers. The goal of this paper is to compute this group and show that when $L$ is a link of at least six components it is nontrivial. 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.