  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.
