A supergroup series for knot complements
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911672493408256 |
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| author | Chae, John |
| author_facet | Chae, John |
| contents | We introduce a three variable series invariant $F_K (y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2|1)$. The invariant is a generalization of the $sl(2|1)$-series invariant $\hat{Z}(q)$ for closed 3-manifolds introduced by Ferrari and Putrov and an extension of the two variable series invariant defined by Gukov and Manolescu (GM) to the Lie superalgebra. We derive a surgery formula relating $F_K (y,z,q)$ to $\hat{Z}(q)$ invariant. We find appropriate expansion chambers for certain infinite families of torus knots and compute explicit examples. Furthermore, we provide evidence for a non semisimple $Spin^c$ decorated TQFT from the three variable series. We observe that the super $F_K (y,z,q)$ itself and its results exhibit distinctive features compared to the GM series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10279 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A supergroup series for knot complements Chae, John Geometric Topology High Energy Physics - Theory Mathematical Physics Quantum Algebra We introduce a three variable series invariant $F_K (y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2|1)$. The invariant is a generalization of the $sl(2|1)$-series invariant $\hat{Z}(q)$ for closed 3-manifolds introduced by Ferrari and Putrov and an extension of the two variable series invariant defined by Gukov and Manolescu (GM) to the Lie superalgebra. We derive a surgery formula relating $F_K (y,z,q)$ to $\hat{Z}(q)$ invariant. We find appropriate expansion chambers for certain infinite families of torus knots and compute explicit examples. Furthermore, we provide evidence for a non semisimple $Spin^c$ decorated TQFT from the three variable series. We observe that the super $F_K (y,z,q)$ itself and its results exhibit distinctive features compared to the GM series. |
| title | A supergroup series for knot complements |
| topic | Geometric Topology High Energy Physics - Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2508.10279 |