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 $\mathbb{Z}[x_1^{\pm 1},\dots ,x_m^{\pm 1}]$ 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 $u{v}^{2k-1}$ 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.

36 pages, 1 figure. v3: Much of the paper is new, inculding the main result. Some material from the old version has moved to a new preprint

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