Generalized Seifert surfaces and signatures of colored links |
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Authors: | David Cimasoni Vincent Florens |
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Affiliation: | Department of Mathematics, University of California Berkeley, 970 Evans Hall, Berkeley, California 94720 ; Departamento Ãlgebra, Geometrã y Topologã, Universidad de Valladolid, Prado de la Magdalena s/n, 47011 Valladolid, Spain |
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Abstract: | In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in . This yields an integral valued function on the -dimensional torus, where is the number of colors of the link. The case corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this -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 -dimensional interpretation and the Atiyah-Singer -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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Keywords: | Colored link Seifert surface Levine-Tristram signature slice genus. |
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