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

TL;DR

This paper establishes a new local-move-identity for Z[t,t^{-1}]-Alexander polynomials of 2-links, extending classical 1-dimensional relations, and explores high-dimensional analogues, revealing limitations and compatibilities in normalization.

Contribution

It introduces a novel local-move-identity for 2-link Alexander polynomials and explores their high-dimensional analogues, connecting these polynomials with the alinking number.

Findings

New local-move-identity for 2-links' Alexander polynomials

Relation between Alexander polynomials and alinking number for 2-links

High-dimensional analogues of the identities and normalization limitations

Abstract

A well-known identity (Alex+) - (Alex-)=(t^{1/2}-t^{-1/2}) (Alex0) holds for three 1-links L+, L-, and L0 which satisfy a famous local-move-relation. We prove a new local-move-identity for the Z[t,t^{-1}]-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 Z[t,t^{-1}]-Alexander polynomials of 2-links and the alinking number of 2-links. We show high dimensional analogues of these results. Furthermore we prove that in the 2-dimensional case we cannot normalize the Z[t,t^{-1}]-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.