Local-move identities for the ℤ[t,t−1]-Alexander polynomials of 2-links, the alinking number, and high-dimensional analogues

Abstract

A well-known identity [Formula: see text] holds for three 1-links [Formula: see text], [Formula: see text], and [Formula: see text] which satisfy a famous local-move relation, where [Formula: see text] becomes the Alexander–Conway polynomial of [Formula: see text] if we let [Formula: see text]. We prove a new local-move identity for the [Formula: see text]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander–Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the [Formula: see text]-Alexander polynomials of 2-links and the alinking number of 2-links. We produce high-dimensional analogues of these results. Furthermore, we prove that in the 2-dimensional case we cannot normalize the [Formula: see text]-Alexander polynomials to be compatible with our identity but that in a high-dimensional case we can do that to be compatible with our new identity.