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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Generalized Seifert surfaces and signatures of colored links
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by David Cimasoni and Vincent Florens PDF
Trans. Amer. Math. Soc. 360 (2008), 1223-1264 Request permission

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
  • 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
  • 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.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1223-1264
  • MSC (2000): Primary 57M25
  • DOI: https://doi.org/10.1090/S0002-9947-07-04176-1
  • MathSciNet review: 2357695