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The first Alexander Z(Z)–modules of surface-links and of virtual links

Akio Kawauchi

Geometry & Topology Monographs 14 (2008) 353–371

DOI: 10.2140/gtm.2008.14.353

arXiv: 0904.1853

Abstract

We characterize the first Alexander []–modules of ribbon surface-links in the 4–sphere fixing the number of components and the total genus, and then the first Alexander []–modules of surface-links in the 4–sphere fixing the number of components. Using the result of ribbon torus-links, we also characterize the first Alexander []–modules of virtual links fixing the number of components. For a general surface-link, an estimate of the total genus is given in terms of the first Alexander []–module. We show a graded structure on the first Alexander []–modules of all surface-links and then a graded structure on the first Alexander []–modules of classical links, surface-links and higher-dimensional manifold-links.

Keywords

Alexander module, surface-link, virtual link, infinite cyclic covering, cokernel-free module, symmetric module

Mathematical Subject Classification

Primary: 57M25

Secondary: 57Q35, 57Q45

References
Publication

Received: 6 October 2005
Revised: 9 March 2007
Accepted: 9 March 2007
Published: 29 April 2008

Authors
Akio Kawauchi
Department of Mathematics
Osaka City University
Sugimoto
Sumiyoshi-ku
Osaka 558-8585
Japan