2018/04/10 by Antonio Campillo, Campillo, A., F. Delgado +3
Mathematics · #14B05 #32S25 #57M25 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1804.03419
openalex publication_date 2018/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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=\(z1,z2,z3)∈ \mathbb C3: z15+z23+z32=0\ with the 5-dimensional sphere \mathbb Sε5=\(z1,z2,z3)∈ \mathbb C3: \vert z1\vert2+\vert z2\vert2+\vert z3\vert2=ε2\. An algebraic link in the Poincaré sphere is the intersection of a germ (C,0)⊂ (S,0) of a complex analytic curve in (S,0) with the sphere \mathbb Sε3 of radius ε small enough. Here we discuss to which extend 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 on (S,0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding E8-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 Poincaré sphere coincides with the Poincaré series of the filtration defined by the corresponding curve valuations. We show that, under conditions similar for those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.