Knots, Perturbative Series and Quantum Modularity
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...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212252 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Knots, Perturbative Series and Quantum Modularity. Stavros Garoufalidis and Don Zagier. SIGMA 20 (2024), 055, 87 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862735217741004800 |
|---|---|
| author | Garoufalidis, Stavros Zagier, Don |
| author_facet | Garoufalidis, Stavros Zagier, Don |
| citation_txt | Knots, Perturbative Series and Quantum Modularity. Stavros Garoufalidis and Don Zagier. SIGMA 20 (2024), 055, 87 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2026-03-21T18:14:23Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212252 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:14:23Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Garoufalidis, Stavros Zagier, Don 2026-02-03T07:54:08Z 2024 Knots, Perturbative Series and Quantum Modularity. Stavros Garoufalidis and Don Zagier. SIGMA 20 (2024), 055, 87 pages 1815-0659 2020 Mathematics Subject Classification: 57N10; 57K16; 57K14; 57K10 arXiv:2111.06645 https://nasplib.isofts.kiev.ua/handle/123456789/212252 https://doi.org/10.3842/SIGMA.2024.055 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. The authors would like to thank the anonymous referees for their extraordinarily careful reading of the manuscript and their detailed suggestions to improve the exposition. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Knots, Perturbative Series and Quantum Modularity Article published earlier |
| spellingShingle | Knots, Perturbative Series and Quantum Modularity Garoufalidis, Stavros Zagier, Don |
| title | Knots, Perturbative Series and Quantum Modularity |
| title_full | Knots, Perturbative Series and Quantum Modularity |
| title_fullStr | Knots, Perturbative Series and Quantum Modularity |
| title_full_unstemmed | Knots, Perturbative Series and Quantum Modularity |
| title_short | Knots, Perturbative Series and Quantum Modularity |
| title_sort | knots, perturbative series and quantum modularity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212252 |
| work_keys_str_mv | AT garoufalidisstavros knotsperturbativeseriesandquantummodularity AT zagierdon knotsperturbativeseriesandquantummodularity |