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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.09269 |
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| _version_ | 1866916434993479680 |
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| author | Spyropoulos, Dean |
| author_facet | Spyropoulos, Dean |
| contents | The Jones-Wenzl projectors are particular elements of the Temperley-Lieb algebra essential to the construction of quantum 3-manifold invariants. As a first step toward categorifying quantum 3-manifold invariants, Cooper and Krushkal categorified these projectors. In another direction, Naisse and Putyra gave a categorification of the Temperley-Lieb algebra compatible with odd Khovanov homology, introducing new machinery called grading categories. We provide a generalization of Naisse and Putyra's work in the spirit of Bar-Natan's canopolies or Jones's planar algebras, replacing grading categories with grading multicategories. We use our setup to prove the existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology. This result quickly implies the existence of a new, "odd" categorification of the colored Jones polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09269 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Jones-Wenzl projectors and odd Khovanov homology Spyropoulos, Dean Geometric Topology Quantum Algebra The Jones-Wenzl projectors are particular elements of the Temperley-Lieb algebra essential to the construction of quantum 3-manifold invariants. As a first step toward categorifying quantum 3-manifold invariants, Cooper and Krushkal categorified these projectors. In another direction, Naisse and Putyra gave a categorification of the Temperley-Lieb algebra compatible with odd Khovanov homology, introducing new machinery called grading categories. We provide a generalization of Naisse and Putyra's work in the spirit of Bar-Natan's canopolies or Jones's planar algebras, replacing grading categories with grading multicategories. We use our setup to prove the existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology. This result quickly implies the existence of a new, "odd" categorification of the colored Jones polynomial. |
| title | Jones-Wenzl projectors and odd Khovanov homology |
| topic | Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2410.09269 |