Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds
We introduce a two-variable refinement Ẑₐ(, ) of plumbed 3-manifold invariants Ẑₐ(), which were previously defined for weakly negative definite plumbed 3-manifolds. We also provide several explicit examples in which we argue the recovering process to obtain Ẑₐ() from Ẑₐ(, ) by taking a limit → 1. F...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211873 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds. Song Jin Ri. SIGMA 19 (2023), 011, 27 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862745367514185728 |
|---|---|
| author | Ri, Song Jin |
| author_facet | Ri, Song Jin |
| citation_txt | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds. Song Jin Ri. SIGMA 19 (2023), 011, 27 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce a two-variable refinement Ẑₐ(, ) of plumbed 3-manifold invariants Ẑₐ(), which were previously defined for weakly negative definite plumbed 3-manifolds. We also provide several explicit examples in which we argue the recovering process to obtain Ẑₐ() from Ẑₐ(, ) by taking a limit → 1. For plumbed 3-manifolds with two high-valency vertices, we analytically compute the limit by using the explicit integer solutions of quadratic Diophantine equations in two variables. Based on numerical computations of the recovered Ẑₐ() for plumbings with two high-valency vertices, we propose a conjecture that the recovered Ẑₐ(), if it exists, is an invariant for all tree plumbed 3-manifolds. Finally, we provide a formula for the Ẑₐ(, ) for the connected sum of plumbed 3-manifolds in terms of those for the components.
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| first_indexed | 2026-04-17T19:00:14Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211873 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T19:00:14Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ri, Song Jin 2026-01-14T09:54:40Z 2023 Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds. Song Jin Ri. SIGMA 19 (2023), 011, 27 pages 1815-0659 2020 Mathematics Subject Classification: 57K31; 57R56; 11D09 arXiv:2205.08197 https://nasplib.isofts.kiev.ua/handle/123456789/211873 https://doi.org/10.3842/SIGMA.2023.011 We introduce a two-variable refinement Ẑₐ(, ) of plumbed 3-manifold invariants Ẑₐ(), which were previously defined for weakly negative definite plumbed 3-manifolds. We also provide several explicit examples in which we argue the recovering process to obtain Ẑₐ() from Ẑₐ(, ) by taking a limit → 1. For plumbed 3-manifolds with two high-valency vertices, we analytically compute the limit by using the explicit integer solutions of quadratic Diophantine equations in two variables. Based on numerical computations of the recovered Ẑₐ() for plumbings with two high-valency vertices, we propose a conjecture that the recovered Ẑₐ(), if it exists, is an invariant for all tree plumbed 3-manifolds. Finally, we provide a formula for the Ẑₐ(, ) for the connected sum of plumbed 3-manifolds in terms of those for the components. I would like to thank my advisor, Pavel Putrov, not only for introducing me to this interesting topic but also for his continuous support and guidance. I would also like to thank the anonymous referees who provided insightful and detailed comments and suggestions on an earlier version of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds Article published earlier |
| spellingShingle | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds Ri, Song Jin |
| title | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds |
| title_full | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds |
| title_fullStr | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds |
| title_full_unstemmed | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds |
| title_short | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds |
| title_sort | refined and generalized ẑ invariants for plumbed 3-manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211873 |
| work_keys_str_mv | AT risongjin refinedandgeneralizedzinvariantsforplumbed3manifolds |