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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.12566 |
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| _version_ | 1866929683890700288 |
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| author | Chae, John |
| author_facet | Chae, John |
| contents | We provide the first computations of colored unknots and Hopf link in $\mathbb{R}\mathbb{P}^3$ using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition of the cotangent bundle of $\mathbb{R}\mathbb{P}^3$ under the large $N$ duality. We find that the link invariants are series in the Kahler parameters of the toric Calabi-Yau manifold and the $q$-expansions of the rational functions of the series have positivity property. We conjecture that they are Poincare series of an infinite dimensional link homology theory for links in $\mathbb{R}\mathbb{P}^3$. We compare our results with that of the $S^3$ and speculate the consequences of the series nature of the invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex Chae, John Mathematical Physics High Energy Physics - Theory Geometric Topology We provide the first computations of colored unknots and Hopf link in $\mathbb{R}\mathbb{P}^3$ using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition of the cotangent bundle of $\mathbb{R}\mathbb{P}^3$ under the large $N$ duality. We find that the link invariants are series in the Kahler parameters of the toric Calabi-Yau manifold and the $q$-expansions of the rational functions of the series have positivity property. We conjecture that they are Poincare series of an infinite dimensional link homology theory for links in $\mathbb{R}\mathbb{P}^3$. We compare our results with that of the $S^3$ and speculate the consequences of the series nature of the invariants. |
| title | Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex |
| topic | Mathematical Physics High Energy Physics - Theory Geometric Topology |
| url | https://arxiv.org/abs/2501.12566 |