Abstract
It is known that the -manifold is diffeomorphic to the complement of the trefoil knot in . E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under . In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.
Citation
W. Duke. Ö. Imamoḡlu. Á. Tóth. "Modular cocycles and linking numbers." Duke Math. J. 166 (6) 1179 - 1210, 15 April 2017. https://doi.org/10.1215/00127094-3793032
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