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15 April 2017 Modular cocycles and linking numbers
W. Duke, Ö. Imamoḡlu, Á. Tóth
Duke Math. J. 166(6): 1179-1210 (15 April 2017). DOI: 10.1215/00127094-3793032

Abstract

It is known that the 3-manifold SL(2,Z)\SL(2,R) is diffeomorphic to the complement of the trefoil knot in S3. 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 SL(2,Z). 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

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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

Information

Received: 3 June 2015; Revised: 8 July 2016; Published: 15 April 2017
First available in Project Euclid: 7 January 2017

zbMATH: 06725012
MathSciNet: MR3635902
Digital Object Identifier: 10.1215/00127094-3793032

Subjects:
Primary: 11F67 , 57M25
Secondary: 11F12 , 11F20

Keywords: Dedekind symbol , linking number , modular forms , modular integrals

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 6 • 15 April 2017
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