A theory of genera for cyclic coverings of links. (English) Zbl 1004.57001

Summary: Following the conceptual analogies between knots and primes, 3-manifolds and number fields, we discuss an analogue in knot theory after the model of the arithmetical theory of genera initiated by Gauss. We present an analog for cyclic coverings of links following along the line of Iyanaga-Tamagawa’s genus theory for cyclic extensions over the rational number field. We also give examples of \(\mathbb{Z}/ 2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\)-coverings of links for which the principal genus theorem does not hold.


57M12 Low-dimensional topology of special (e.g., branched) coverings
11R32 Galois theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions
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