Strong periodicity of links and the coefficients of the Conway polynomial

• Published in: Proceedings of the American Mathematical Society Vol. 136; no. 6; pp. 2217 - 2224
• Main Author: Chbili, Nafaa
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-06-2008
• Subjects: Algebra; Coefficients; Exact sciences and technology; General mathematics; General, history and biography; Geometric topology; Global analysis, analysis on manifolds; Group theory; Group theory and generalizations; Integers; Manifolds and cell complexes; Mathematical induction; Mathematical theorems; Mathematics; Polynomials; Quotients; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Strongly periodic links; equivariant crossing change; Conway polynomial; Circle; Action; Zero; Cyclic group; Periodicity; Homology; Mathematics; Sphere; Fix point; Finite group; Three manifold; Surgery; Ordered group; Behavior; Integral manifold; Quotient; Link; Fixed point
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-08-09266-6

Przytycki and Sokolov proved that a three-manifold admits a semi-free action of the finite cyclic group of order p with a circle as the set of fixed points if and only if M is obtained from the three-sphere by surgery along a strongly p-periodic link L. Moreover, if the quotient three-manifold is an integral homology sphere, then we may assume that L is orbitally separated. This paper studies the behavior of the coefficients of the Conway polynomial of such a link. Namely, we prove that if L is a strongly p-periodic orbitally separated link and p is an odd prime, then the coefficient a_{2i}(L) is congruent to zero modulo p for all i such that 2i<p-1.