Algebraic links in the Poincar\'e sphere and the Alexander polynomials

Abstract

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincar\'e 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 Poincar\'e sphere is the intersection of a germ $(C,0)\subset (S,0)$ of a complex analytic curve in $(S,0)$ with the sphere ${\mathbb S}_{\varepsilon}^3$ of radius $\varepsilon$ small enough. Here we discuss to which extend the Alexander polynomial in several variables of an algebraic link in the Poincar\'e sphere determines the topology of the link. We show that, if the strict transform of a curve on $(S,0)$ 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. Alexander polynomial of an algebraic link in the Poincar\'e sphere coincides with the Poincar\'e series of the filtration defined by the corresponding curve valuations. We show that, under conditions similar for those for curves, the Poincar\'e series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.