Temperley-Lieb Categories on Non-Orientable Surfaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918061495287808 |
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| author | Ibarra, Dionne Montoya-Vega, Gabriel Morris, Benjamin |
| author_facet | Ibarra, Dionne Montoya-Vega, Gabriel Morris, Benjamin |
| contents | In this paper we present the construction of a skeletal diagram category, which we call the square with bands category. This category extends the Temperley-Lieb (TL) category, where morphisms now include diagrams of embedded curves on (possibly) non-orientable bounded surfaces, and involves three parameters associated to simple closed curves. Such diagrams utilise handle decompositions for surfaces and are considered up to a handle slide equivalence. We define a tensor product on this category, extending the well-known tensor product on the TL category, and a full set of monoidal generators is given, which includes the TL generators, a family of orientable genus one diagrams, and a family of non-orientable diagrams.
This document constitutes an initial draft of ongoing research with preliminary reporting of some results in the last section; a subsequent version including a detailed introduction and full proofs will follow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Temperley-Lieb Categories on Non-Orientable Surfaces Ibarra, Dionne Montoya-Vega, Gabriel Morris, Benjamin Category Theory Quantum Algebra Primary: 18M05, Secondary: 57K20, 18F99 In this paper we present the construction of a skeletal diagram category, which we call the square with bands category. This category extends the Temperley-Lieb (TL) category, where morphisms now include diagrams of embedded curves on (possibly) non-orientable bounded surfaces, and involves three parameters associated to simple closed curves. Such diagrams utilise handle decompositions for surfaces and are considered up to a handle slide equivalence. We define a tensor product on this category, extending the well-known tensor product on the TL category, and a full set of monoidal generators is given, which includes the TL generators, a family of orientable genus one diagrams, and a family of non-orientable diagrams. This document constitutes an initial draft of ongoing research with preliminary reporting of some results in the last section; a subsequent version including a detailed introduction and full proofs will follow. |
| title | Temperley-Lieb Categories on Non-Orientable Surfaces |
| topic | Category Theory Quantum Algebra Primary: 18M05, Secondary: 57K20, 18F99 |
| url | https://arxiv.org/abs/2506.14319 |