Two-variable Conway polynomial and Cochran's derived invariants

We note that the Conway potential function ${\mathrm{\Omega }}_L$ of an $m$-component link $L$, $m>1$, can be expressed as ${\mathrm{\Omega }}_L(x_1,\dots ,x_m)={\mathrm{\Theta }}_L({\mathrm{\nabla }}_L(x_1-x_1^{-1},\dots ,x_m-x_m^{-1}))$ for a unique ${\mathrm{\nabla }}_L\in \mathbb{Z}[z_1,\dots ,z_m]$, where ${\mathrm{\Theta }}_L$ is a certain endomorphism of the additive group of $\mathbb Z[x_1^{\pm1},\dots,x_m^{\pm1}]$ which depends only on the pairwise linking numbers of the components of $L$. Motivated by applications to topological isotopy, we study the formal power series ${\bar{\mathrm{\nabla }}}_L$, obtained by dividing ${\mathrm{\nabla }}_L$ by the Conway polynomials of the components of $L$. For a $2$-component link with $lk(L)=0$, the coefficient ${\alpha }_{1,2k-1}$ of ${\bar{\mathrm{\nabla }}}_L(u,v)$ at $u{v}^{2k-1}$ equals Cochran’s derived invariant $(-1{)}^{k+1}{\beta }^k(L)$. While this can be deduced from a result of G.-T. Jin, which he proved using the surgical view of the Alexander polynomial, we provide an alternative proof, using Seifert matrices. Our main result is a formula for the same coefficient ${\alpha }_{1,2k-1}$ in the geometrically subtler case $lk(L)=1$. Namely we express it in terms of generalized Cochran invariants ${\beta }_F^{ij}(P,Q)$, which were studied by Gilmer–Livingston (when $P=Q$) and by Tsukamoto–Yasuhara (when $j=0$) and are closely related to the Cochran pairing in the infinite cyclic covering of a knot.