Higher Rank Ẑ and FK
We study q-series-valued invariants of 3-manifolds that depend on the choice of a root system . This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057], where they focused on the = SU(2) case. Although a full mathemati...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
| Main Author: | |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210706 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Higher Rank Ẑ and FK. Sunghyuk Park. SIGMA 16 (2020), 044, 17 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862742065283072000 |
|---|---|
| author | Park, Sunghyuk |
| author_facet | Park, Sunghyuk |
| citation_txt | Higher Rank Ẑ and FK. Sunghyuk Park. SIGMA 16 (2020), 044, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study q-series-valued invariants of 3-manifolds that depend on the choice of a root system . This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057], where they focused on the = SU(2) case. Although a full mathematical definition for these ''invariants'' is lacking yet, we define Ẑ for negative definite plumbed 3-manifolds and FGK for torus knot complements. As in the = SU(2) case by Gukov and Manolescu, there is a surgery formula relating FGK to Ẑ of a Dehn surgery on the knot K. Furthermore, specializing to symmetric representations, FGK satisfies a recurrence relation given by the quantum A-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.
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| first_indexed | 2025-12-17T12:04:33Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210706 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:33Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Park, Sunghyuk 2025-12-15T15:25:29Z 2020 Higher Rank Ẑ and FK. Sunghyuk Park. SIGMA 16 (2020), 044, 17 pages 1815-0659 2020 Mathematics Subject Classification: 57K16; 57K31; 81R50 arXiv:1909.13002 https://nasplib.isofts.kiev.ua/handle/123456789/210706 https://doi.org/10.3842/SIGMA.2020.044 We study q-series-valued invariants of 3-manifolds that depend on the choice of a root system . This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057], where they focused on the = SU(2) case. Although a full mathematical definition for these ''invariants'' is lacking yet, we define Ẑ for negative definite plumbed 3-manifolds and FGK for torus knot complements. As in the = SU(2) case by Gukov and Manolescu, there is a surgery formula relating FGK to Ẑ of a Dehn surgery on the knot K. Furthermore, specializing to symmetric representations, FGK satisfies a recurrence relation given by the quantum A-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants. I would like to thank my advisor Sergei Gukov for his invaluable guidance, as well as Francesca Ferrari, Sarah Harrison, Ciprian Manolescu, and Nikita Sopenko for helpful conversations. Special thanks go to Nikita Sopenko for his kind help with Mathematica coding. I would also like to thank the anonymous referees for their useful comments that helped to improve the paper. The author was supported by the Kwanjeong Educational Foundation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Higher Rank Ẑ and FK Article published earlier |
| spellingShingle | Higher Rank Ẑ and FK Park, Sunghyuk |
| title | Higher Rank Ẑ and FK |
| title_full | Higher Rank Ẑ and FK |
| title_fullStr | Higher Rank Ẑ and FK |
| title_full_unstemmed | Higher Rank Ẑ and FK |
| title_short | Higher Rank Ẑ and FK |
| title_sort | higher rank ẑ and fk |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210706 |
| work_keys_str_mv | AT parksunghyuk higherrankzandfk |