Two-variable Conway polynomial and Cochran's derived invariants

Abstract

We note that the Conway potential function $\Omega_L$ of an $m$-component link $L$, $m>1$, can be expressed as $\Omega_L(x_1,\dots,x_m)=\Theta_L(\nabla_L(x_1-x_1^{-1},\dots,x_m-x_m^{-1}))$ for a unique $\nabla_L\in\mathbb Z[z_1,\dots,z_m]$, where $\Theta_L$ 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 $\bar\nabla_L$, obtained by dividing $\nabla_L$ 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 $\bar\nabla_L(u,v)$ at $uv^{2k-1}$ 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.