Are algebraic links in the Poincaré sphere determined by their Alexander polynomials? (Q2287998)

An algebraic link in the \(3\)-sphere is the intersection of a plane curve \((C,0) \subset (\mathbb C^2,0)\) defined by a complex analytic function and the \(3\)-sphere with sufficiently small radius. For an algebraic link, \textit{M. Yamamoto} [Topology 23, 277--287 (1984; Zbl 0581.57002)] proved that the Alexander polynomial in several variables (corresponding to the components) determines the topological type of the link. The authors of the article under review discuss the same problem for the algebraic links in the Poincaré sphere, namely, the intersection of the \(E_8\)-singularity \(S=\{(z_1,z_2,z_3) \in \mathbb C^3 : z_1^5+z_2^3+z_3^2=0\}\) with the \(5\)-sphere \(\{(z_1,z_2,z_3) \in \mathbb C^3 : |z_1|^2+|z_2|^2+|z_3|^2=\epsilon^2\}\) with sufficiently small \(\epsilon>0\). Note that the germ of every complex curve on \(S\) is a hypersurface because the \(E_8\)-singularity is factorial, and that the topological type of an algebraic link is equivalent to the combinatorial type of the minimal embedded resolution of the curve singularity. The authors prove that if the strict transform of a complex curve in \((S,0)\) does not intersect the component of the exceptional prime divisor corresponding to the end of the longest tail of the \(E_8\)-diagram, then the Poincaré series of the filtration defined by the corresponding curve valuations determines the combinatorial type of the minimal embedded resolution of the curve. In fact, the authors have proved that the Alexander polynomial determines the Poincaré series [Comment. Math. Helv. 80, No. 1, 95--102 (2005; Zbl 1075.14024)]. They also prove that, under similar conditions, the Poincaré series of a collection of divisorial valuations (corresponding prime exceptional divisors) determines the combinatorial type of the minimal resolution of the collection.