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Generalized Seifert surfaces and signatures of colored links. (English) Zbl 1132.57004
Let \(L\) be a link and \(A_{L}(t)\) a presentation matrix for the Alexander module of L. Let \(\omega \in \mathbb C, | \omega| = 1, \omega \neq 1\), then the signature \(\sigma_L\) and nullity \(\eta_L\) of \(H(\omega) = (1-\overline{\omega})A(\omega)\) define 2 functions, \(\sigma_{L}(\omega)\) and \(\eta_{L}(\omega)\) from \(S^1-{1} \to \mathbb Z\) called the Levine-Tristram signature.
The authors generalize the Levine-Tristram signature to \(\mu\)-colored links and the consequent signature and nullity of a matrix evaluated at points of \([S^1-{1}]^\mu\).Various properties of the Levine-Tristram signature which carry over to their generalization are proved.
Their invariant is so under colored concordance, and yields a lower bound for the slice genus.
The same generalization in the special case of \(\mu = 2 \) for 2-component links was investigated by D. Cooper in his 1982 Univ. of Warwick PhD thesis and in [Proc. Conf., Bangor/Engl. 1979, Vol. 1, Lond. Math. Soc. Lect. Note Ser. 48, 51–66 (1982; Zbl 0483.57004)].

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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