Kirby Problem 1.8

Definitions and mathematical statement

Let $Y$ be an oriented integral homology $3$-sphere, meaning $H_*(Y;\mathbb Z)\cong H_*(S^3;\mathbb Z)$, and let $L=L_1\cup\cdots\cup L_m$ be an ordered oriented link in $Y$. Put $X=Y\setminus\operatorname{int}\nu L$. Sending the $m$ oriented meridians to the standard basis gives $\pi_1(X)\twoheadrightarrow\mathbb Z^m$ and a corresponding regular cover $\widetilde X$. Its Alexander module $H_1(\widetilde X;\mathbb Z)$ is a module over $R_m=\mathbb Z[t_1^{\pm1},\ldots,t_m^{\pm1}]$ by deck transformations.

The multivariable Alexander polynomial $\Delta_L$ is the order of this module: the greatest common divisor of the maximal minors in a finite presentation, with value zero if the module has positive rank. It is defined up to a unit $\pm t_1^{a_1}\cdots t_m^{a_m}$; the zero module has order one.

For each $m$, characterize precisely the Laurent polynomials that occur as $\Delta_L$ for links in $S^3$. Ask the same realization question when integral homology spheres are allowed as ambient manifolds. A characterization must give both necessary conditions and realizations for every polynomial satisfying them, respecting the stated unit ambiguity.

Short English statement

Which multivariable Laurent polynomials are Alexander polynomials of links in a sphere or a homology sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2667 (KP-1.8), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.8. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.