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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalized Seifert surfaces and signatures of colored links


Authors: David Cimasoni and Vincent Florens
Journal: Trans. Amer. Math. Soc. 360 (2008), 1223-1264
MSC (2000): Primary 57M25
Published electronically: October 23, 2007
MathSciNet review: 2357695
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Abstract: In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in $ S^3$. This yields an integral valued function on the $ \mu$-dimensional torus, where $ \mu$ is the number of colors of the link. The case $ \mu=1$ corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this $ \mu$-variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a $ 4$-dimensional interpretation and the Atiyah-Singer $ G$-signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.


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Additional Information

David Cimasoni
Affiliation: Department of Mathematics, University of California Berkeley, 970 Evans Hall, Berkeley, California 94720
Email: cimasoni@math.berkeley.edu

Vincent Florens
Affiliation: Departamento Ãlgebra, Geometrã y Topologã, Universidad de Valladolid, Prado de la Magdalena s/n, 47011 Valladolid, Spain
Address at time of publication: Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case Postale 64, 1211 Genève 4, Switzerland
Email: vincent_florens@yahoo.fr, vincent.florens@math.unige.ch

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04176-1
PII: S 0002-9947(07)04176-1
Keywords: Colored link, Seifert surface, Levine-Tristram signature, slice genus.
Received by editor(s): May 6, 2005
Received by editor(s) in revised form: August 23, 2005
Published electronically: October 23, 2007
Additional Notes: The first author was supported by the Swiss National Science Foundation.
The second author was supported by Marie-Curie, MCHF-2001-0615.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.