Volumes of Hyperbolic Three-Manifolds Associated with Modular Links
Abstract
:1. Introduction
1.1. Knots Associated with Closed Geodesics
1.2. Modular Links and Their Volumes
1.3. Results and Structure of This Paper
2. Modular Links and Quadratic Fields
2.1. Periodic Geodesics on the Modular Surface
2.2. Quadratic Forms and Geodesics
3. Coding of Geodesics
3.1. Conjugacy Classes in and Words in Two Generators
3.2. From Matrices to Words
3.3. Curves from Words and the Template
4. Computational Method
- The fundamental unit has norm . In this case, is the regulator and the narrow class number is twice the class number. Thus, the sum of the geometric lengths is given by .
- The fundamental unit has norm . In this case, the unit we use is not the fundamental unit, but . i.e., its log is twice the regulator. On the other hand, the narrow class number is in this case the same as the class number, and thus the sum of the geometric lengths is given by in this case as well.
5. Results and Discussion
Author Contributions
Funding
Conflicts of Interest
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Brandts, A.; Pinsky, T.; Silberman, L. Volumes of Hyperbolic Three-Manifolds Associated with Modular Links. Symmetry 2019, 11, 1206. https://doi.org/10.3390/sym11101206
Brandts A, Pinsky T, Silberman L. Volumes of Hyperbolic Three-Manifolds Associated with Modular Links. Symmetry. 2019; 11(10):1206. https://doi.org/10.3390/sym11101206
Chicago/Turabian StyleBrandts, Alex, Tali Pinsky, and Lior Silberman. 2019. "Volumes of Hyperbolic Three-Manifolds Associated with Modular Links" Symmetry 11, no. 10: 1206. https://doi.org/10.3390/sym11101206