Duke Mathematical Journal

Modular cocycles and linking numbers

W. Duke, Ö. Imamoḡlu, and Á. Tóth

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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.

Article information

Source
Duke Math. J., Volume 166, Number 6 (2017), 1179-1210.

Dates
Received: 3 June 2015
Revised: 8 July 2016
First available in Project Euclid: 7 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1483758030

Digital Object Identifier
doi:10.1215/00127094-3793032

Mathematical Reviews number (MathSciNet)
MR3635902

Zentralblatt MATH identifier
06725012

Subjects
Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 11F12: Automorphic forms, one variable 11F20: Dedekind eta function, Dedekind sums

Keywords
Modular forms modular integrals Dedekind symbol linking number

Citation

Duke, W.; Imamoḡlu, Ö.; Tóth, Á. Modular cocycles and linking numbers. Duke Math. J. 166 (2017), no. 6, 1179--1210. doi:10.1215/00127094-3793032. https://projecteuclid.org/euclid.dmj/1483758030


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