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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
MR Author ID: 677173

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

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.