Hyperbolic 3-manifolds and Cluster Algebras

Abstract

We advocate the use of cluster algebras and their y-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster y-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.

v2: 26 pages, 17 figures

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The quantum dilogarithm and representations quantum cluster varieties
• R-Twisting and 4d/2d Correspondences
• Q-systems as cluster algebras
• Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations
• 3d N=2 Theories from Cluster Algebras

Cited by in corpus (9)

• Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus
• Network, Cluster coordinates and N=2 theory I
• Quantum Dilogarithm Identities at Root of Unity
• Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Knots
• Note on character varieties and cluster algebras
• Invariant Functions On Cluster Ensembles
• Braiding Operator via Quantum Cluster Algebra
• Jacobian matrices of Y-seed mutations
• Torsion functions on moduli spaces in view of the cluster algebra