Knots and Their Related -Series
We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane, which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that define a PSL₂(ℤ)-cocycle on the space of matrix-value...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212049 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Knots and Their Related -Series. Stavros Garoufalidis and Don Zagier. SIGMA 19 (2023), 082, 39 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Резюме: | We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane, which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that define a PSL₂(ℤ)-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.
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| ISSN: | 1815-0659 |