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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
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| 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/123456789/211873 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Refined and Generalized Ẑ Invariants for Plumbed 3-Manifolds. Song Jin Ri. SIGMA 19 (2023), 011, 27 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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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| ISSN: | 1815-0659 |