Knots, Perturbative Series and Quantum Modularity

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2024
• Hauptverfasser: Garoufalidis, Stavros, Zagier, Don
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2024
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/212252
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: 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) entry, where σ₀ is the trivial and σ₁ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity e²πⁱα as an element of the Habiro ring, and the remaining entries belong to generalized Habiro rings of number fields; (b) the first column is given by the perturbative power series of Dimofte-Garoufalidis; (c) the columns of are fundamental solutions of a linear -difference equation; (d) the matrix defines an SL₂(ℤ)-cocycle Wᵧ in matrix-valued functions on ℚ that conjecturally extends to a smooth function on ℝ and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series () to actual functions. The two invariants and Wᵧ are related by a refined quantum modularity conjecture, which we illustrate in detail for the three simplest hyperbolic knots, the 4₁, 5₂, and (−2, 3, 7) pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent -series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.