Embedded surface invariants via the Broda-Petit construction
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915604797063168 |
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| author | Lee, Ik Jae Yetter, David N |
| author_facet | Lee, Ik Jae Yetter, David N |
| contents | We recall Petit's construction of "dichromatic" invariants of 4-manifolds computed from Kirby diagrams using a nested pair of ribbon fusion categories $ B \subset C $ as initial data. Along the way we prove a lemma that fits the use of formal linear combinations of simple objects with quantum dimensions a coefficients as in the constructions of Reshetikhin-Turaev, Broda, and Petit more firmly in the functorial framework favored by the authors.
We then show that Hughes et al.'s banded-link presentations of surfaces embedded in 4-manifolds provide a means whereby Frobenius algebra in $B$ together with a suitable module over it lying in $C$, give rise to an invariant of a surface-4-manifold pair. We provide a class of examples of suitable initial data and compute sufficient examples to show the invariant is sensitive to both genus and knotting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_11380 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Embedded surface invariants via the Broda-Petit construction Lee, Ik Jae Yetter, David N Quantum Algebra Geometric Topology 57K45, 57R56, 57K41 We recall Petit's construction of "dichromatic" invariants of 4-manifolds computed from Kirby diagrams using a nested pair of ribbon fusion categories $ B \subset C $ as initial data. Along the way we prove a lemma that fits the use of formal linear combinations of simple objects with quantum dimensions a coefficients as in the constructions of Reshetikhin-Turaev, Broda, and Petit more firmly in the functorial framework favored by the authors. We then show that Hughes et al.'s banded-link presentations of surfaces embedded in 4-manifolds provide a means whereby Frobenius algebra in $B$ together with a suitable module over it lying in $C$, give rise to an invariant of a surface-4-manifold pair. We provide a class of examples of suitable initial data and compute sufficient examples to show the invariant is sensitive to both genus and knotting. |
| title | Embedded surface invariants via the Broda-Petit construction |
| topic | Quantum Algebra Geometric Topology 57K45, 57R56, 57K41 |
| url | https://arxiv.org/abs/2303.11380 |