Integer Milnor coefficients and closed-link residues

Assume AC for the string-link quotient branch. Use the closed-link residue definition one finite maximum index length at a time, so its based word-lift data are finite.

Let $\tau$ be a based $r$-component string link with $r\ge 1$ and bottom meridian symbol group $F_r=⟨x_1,\dots ,x_r⟩$, exterior group $G={\pi }_1(X_{\tau })$, and based string-link longitudes ${\lambda }_1,\dots ,{\lambda }_r$. With compatible stem accesses this free group identifies the full endpoint fundamental group. For arbitrary independently based meridians, use instead its meridian map on each finite nilpotent quotient, which is still an isomorphism by the end-identification theorem; no full-group free-basis assertion is made. Longitudes retain their specified parallel and access data. For a multi-index $I=(i_1,\dots ,i_k,j)$ with $k\ge 1$ and each index in $\{1,\dots ,r\}$, let $w_{j,k}\in F_r\mathrm{/}{\mathrm{\Gamma }}_{k+1}{F}_r$ be the unique class whose image under the bottom-end nilpotent isomorphism is the class of ${\lambda }_j$ in $G\mathrm{/}{\mathrm{\Gamma }}_{k+1}G$. Define the integer coefficient ${\mu }_{\tau }(I)=[X_{i_1}\cdots X_{i_k}]E(w_{j,k})\in \mathbb{Z},$ where the bracket means the coefficient of that word in the finite Magnus expansion through degree $k$. The commutator-degree lemma makes this coefficient independent of the representative of $w_{j,k}$.

For a closed labeled link $L$, choose a common basing and its preferred zero-self-meridian longitudes. For every component $j$ and every finite $q\ge 2$, choose a lift $w_{j,q}\in F_r\mathrm{/}{\mathrm{\Gamma }}_q{F}_r$ of the class of its longitude in $G\mathrm{/}{\mathrm{\Gamma }}_{q}G$, where $G={\pi }_1(E_L)$ and $F_r$ is free on the component meridian symbols. Such lifts exist because those meridians generate every finite nilpotent quotient. For $I=(i_1,\dots ,i_k,j)$, define the based coefficient ${\mu }_L^b(I)$ by taking the coefficient of $X_{i_1}\cdots X_{i_k}$ in $E(w_{j,k+1})$ through degree $k$. Let ${\mathrm{\Delta }}_I^b$ be the nonnegative greatest common divisor of the finite set of all ${\mu }_L^b(J)$ for sequences $J$ of lengths from $2$ to $\mathrm{\mid }I\mathrm{\mid }-1$ that occur as subsequences of cyclic permutations of $I$. The gcd of an empty set is $0$. Define the based residue candidate ${\bar{\mu }}_L^b(I):={\mu }_L^b(I)(\mathrm{m}\mathrm{o}\mathrm{d}{\mathrm{\Delta }}_I^b)\in \mathbb{Z}\mathrm{/}{\mathrm{\Delta }}_I^b\mathbb{Z},$ with $\mathbb{Z}\mathrm{/}0\mathbb{Z}=\mathbb{Z}$. Independence of the basing and word lifts is not part of this definition; it is a separate theorem. A based Borromean example with $I=(1,2,3)$ uses this convention.

Lower-central quotients and finite Magnus expansion

Based string-link longitudes and closed-link peripheral data

Representative-independent truncated coefficients

End isomorphisms on finite lower-central quotients

Component meridians generate finite closed-link nilpotent quotients

Existence of preferred zero-self-meridian longitudes

The length of $I$ is $k+1$; its final index names the longitude component, and the preceding indices name the Magnus word. The gcd convention records exactly the lower-order data used in the classical residue definition. The isotopy theorem later proves that the based residue candidate is independent of its choices.