Knots, Perturbative Series and Quantum Modularity

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Authors: Garoufalidis, Stavros, Zagier, Don
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212252
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Knots, Perturbative Series and Quantum Modularity. Stavros Garoufalidis and Don Zagier. SIGMA 20 (2024), 055, 87 pages

We introduce an invariant of a hyperbolic knot, which is a map α ↦ α() from ℚ/ℤ to matrices with entries in ℚ¯[[]] and with rows and columns indexed by the boundary parabolic SL2(ℂ) representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their (σ₀, σ₁...