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Are algebraic links in the Poincaré sphere determined by their Alexander polynomials?статья
Информация о цитировании статьи получена из
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Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 24 января 2020 г.
- Авторы: Campillo A., Delgado F., Gusein-Zade S.M.
- Журнал: Mathematische Zeitschrift
- Том: 294
- Номер: 1-2
- Год издания: 2020
- Издательство: Springer Verlag
- Местоположение издательства: Germany
- Первая страница: 593
- Последняя страница: 613
- DOI: 10.1007/s00209-019-02282-0
- Аннотация: 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={(z_1,z_2,z_3)\in\C^3: z_1^5+z_2^3+z_3^2=0} with the 5-dimensional sphere S^5={(z_1,z_2,z_3)\in\C^3: |z_1|^2+|z_2|^2+|z_3|^2=1}. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S,0) with the sphere S_{\eps}^5 of radius \eps small enough. Here we discuss to which extent 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 in (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. 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 coincides 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.
- Добавил в систему: Гусейн-Заде Сабир Меджидович