A theory of genera for cyclic coverings of links

Authors :: Masanori Morishita
Source :: Proc. Japan Acad. Ser. A Math. Sci. 77, no. 7 (2001), 115-118
Publication Year :: 2001
Publisher :: The Japan Academy, 2001.
Abstract: 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 extentions over the rational number field. We also give examples of $\mathbf{Z} / 2\mathbf{Z} \times \mathbf{Z} / 2\mathbf{Z}$-coverings of links for which the principal genus theorem does not hold.