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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212049 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Knots and Their Related -Series. Stavros Garoufalidis and Don Zagier. SIGMA 19 (2023), 082, 39 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862706554789167104 |
|---|---|
| author | Garoufalidis, Stavros Zagier, Don |
| author_facet | Garoufalidis, Stavros Zagier, Don |
| citation_txt | Knots and Their Related -Series. Stavros Garoufalidis and Don Zagier. SIGMA 19 (2023), 082, 39 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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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| first_indexed | 2026-03-19T02:17:01Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212049 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T02:17:01Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Garoufalidis, Stavros Zagier, Don 2026-01-23T10:12:48Z 2023 Knots and Their Related -Series. Stavros Garoufalidis and Don Zagier. SIGMA 19 (2023), 082, 39 pages 1815-0659 2020 Mathematics Subject Classification: 57N10; 57K16; 57K14; 57K10 arXiv:2304.09377 https://nasplib.isofts.kiev.ua/handle/123456789/212049 https://doi.org/10.3842/SIGMA.2023.082 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. The authors would like to thank Tudor Dimofte, Gie Gu, Rinat Kashaev, Marcos Marino, and Sander Zwegers for their input and the anonymous referees for a careful reading of the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Knots and Their Related -Series Article published earlier |
| spellingShingle | Knots and Their Related -Series Garoufalidis, Stavros Zagier, Don |
| title | Knots and Their Related -Series |
| title_full | Knots and Their Related -Series |
| title_fullStr | Knots and Their Related -Series |
| title_full_unstemmed | Knots and Their Related -Series |
| title_short | Knots and Their Related -Series |
| title_sort | knots and their related -series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212049 |
| work_keys_str_mv | AT garoufalidisstavros knotsandtheirrelatedseries AT zagierdon knotsandtheirrelatedseries |