On the Characterization Problem of Alexander Polynomials of Closed 3-Manifolds

TL;DR

This paper characterizes Alexander polynomials of closed orientable 3-manifolds with first Betti number 1 and provides partial results for higher Betti numbers, revealing which symmetric polynomials can occur as Alexander polynomials.

Contribution

It extends Levine's theorem to characterize Alexander polynomials for certain 3-manifolds and identifies conditions under which symmetric polynomials are realizable.

Findings

Symmetric polynomials with non 0 trace are Alexander polynomials for Betti number 1, 2, or 3.

For Betti number 1, only symmetric polynomials with non 0 trace can occur.

Alexander polynomial cannot be 1 for Betti number > 3.

Abstract

We give a characterization for the Alexander Polynomials of closed orientable 3-manifolds M with first Betti number 1, as well as some partial results for the characterization problem for M having first Betti number > 1. We first prove an analogue of a theorem of Levine: that the product of an Alexander polynomial of M with a symmetric polynomial in the same number of variables having non 0 trace, is again an Alexander polynomial of a closed orientable 3-manifold. Using the fact that there exists M with Alexander polynomial = 1 for M with first Betti number 1, 2 or 3, we conclude that symmetric polynomials of non 0 trace in 1, 2 or 3 variables are Alexander polynomials of closed orientable 3-manifolds. When the first Betti number of M is 1 we prove that non 0 trace symmetric polynomials are the only ones that can arise. Finally, for M with first Betti number > 3 we prove that the Alexander polynomial can not be 1, implying that for such manifolds not all symmetric polynomials having non 0 trace will occur.