Are algebraic links in the Poincaré sphere determined by their Alexander polynomials?

Abstract

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface $S=\{(z_1,z_2,z_3)\in {\mathbb {C}}^3: z_1^5+z_2^3+z_3^2=0\}$ with the 5-dimensional sphere ${\mathbb {S}}_{\varepsilon }^5=\{(z_1,z_2,z_3)\in {\mathbb {C}}^3: \vert z_1\vert ^2+\vert z_2\vert ^2+\vert z_3\vert ^2=\varepsilon ^2\}$. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S, 0) with the sphere ${\mathbb S}^5_\varepsilon$ of radius $\varepsilon$ small enough. Here we discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. We show that, if the strict transform of a curve in (S, 0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding $E_8$-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. (They coincide with each other for a reducible curve singularity and differ by the factor $(1-t)$ for an irreducible one.) We show that, under conditions similar to those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.