RT info:eu-repo/semantics/article
T1 Are algebraic links in the Poincaré sphere determined by their Alexander polynomials?
A1 Campillo López, Antonio
A1 Delgado de la Mata, Félix
A1 Gusein-Zade, Sabir M.
K1 Algebraic links
K1 Conexiones algebráicas
K1 Poincaré sphere
K1 Esfera de Poincaré
K1 Alexander polynomial
K1 Polinomio de Alexander
AB 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)∈C3:z51+z32+z23=0} with the 5-dimensional sphere S5ε={(z1,z2,z3)∈C3:|z1|2+|z2|2+|z3|2=ε2}. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S, 0) with the sphere S5ε of radius ε 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 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. 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.
PB Springer
SN 1432-1823
YR 2020
FD 2020
LK http://uvadoc.uva.es/handle/10324/40736
UL http://uvadoc.uva.es/handle/10324/40736
LA eng
NO Mathematische Zeitschrift, 2020, vol. 294. p. 593-613
NO Producción Científica
DS UVaDOC
RD 26-abr-2024