Linking Numbers of Modular Knots

§1. In this note we examine some questions about closed geodesics on the modular surface X which are suggested by the spectacular pictures from Ghys’ talk [G] as well as his paper with Leys [G-L]. The discussion below follows closely my letter to Mozzochi [Sa1] and as is done there we will only outline the proofs of the main results. Detailed proofs of the delicate estimates that are needed will appear in Mozzochi’s article [Mo].

[1]  E. Grosswald.  Dedekind-Rademacher Sums , 1971 .

[2]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[3]  Prime Geodesic Theorems , 1980 .

[4]  D. Hejhal.  The Selberg trace formula for PSL (2, IR) , 1983 .

[5]  Kevin Walker,et al.  A. Dedekind Sums , 1992 .

[6]  Elie Bienenstock,et al.  Neural Networks and the Bias/Variance Dilemma , 1992, Neural Computation.

[7]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[8]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[9]  Igor Rivin.  Growth in free groups (and other stories) , 1999 .

[10]  Felipe Cucker,et al.  Best Choices for Regularization Parameters in Learning Theory: On the Bias—Variance Problem , 2002, Found. Comput. Math..

[11]  Van,et al.  A gene-expression signature as a predictor of survival in breast cancer. , 2002, The New England journal of medicine.

[12]  S. Horvath,et al.  A General Framework for Weighted Gene Co-Expression Network Analysis , 2005, Statistical applications in genetics and molecular biology.

[13]  Emmanuel Barillot,et al.  Classification of microarray data using gene networks , 2007, BMC Bioinformatics.

[14]  É. Ghys.  Knots and dynamics , 2006 .

[15]  Jeffrey T. Chang,et al.  Oncogenic pathway signatures in human cancers as a guide to targeted therapies , 2006, Nature.

[16]  Inderjit S. Dhillon,et al.  Weighted Graph Cuts without Eigenvectors A Multilevel Approach , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  T. Ideker,et al.  Network-based classification of breast cancer metastasis , 2007, Molecular systems biology.

[18]  Ian M. Donaldson,et al.  iRefIndex: A consolidated protein interaction database with provenance , 2008, BMC Bioinformatics.

[19]  Doheon Lee,et al.  Inferring Pathway Activity toward Precise Disease Classification , 2008, PLoS Comput. Biol..

[20]  Shu Yang,et al.  Target Detection Via Network Filtering , 2009, IEEE Transactions on Information Theory.

[21]  K. Soundararajan,et al.  The prime geodesic theorem , 2010, 1011.5486.