Fusion Trees and Homological Representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908622529757184 |
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| author | Kim, Sung |
| author_facet | Kim, Sung |
| contents | We establish an identification between the spaces of $α$-fusion trees in non-semisimple topological quantum computation (NSS TQC) and a family of homological representations of the braid group known as the Lawrence representations specialized at roots of unity. Leveraging this connection, we provide a new proof of Ito's colored Alexander invariant formula using graphical calculus. Inspired by Anghel's topological model, we derive a formula involving the Hermitian pairing of fusion trees. This formula verifies that non-semisimple quantum knot invariants can be explicitly encoded via the language of fusion trees in the NSS TQC mathematical architecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_27218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fusion Trees and Homological Representations Kim, Sung Geometric Topology Quantum Algebra 18M15, 18M20, 18M30, 57K16, 81R50 We establish an identification between the spaces of $α$-fusion trees in non-semisimple topological quantum computation (NSS TQC) and a family of homological representations of the braid group known as the Lawrence representations specialized at roots of unity. Leveraging this connection, we provide a new proof of Ito's colored Alexander invariant formula using graphical calculus. Inspired by Anghel's topological model, we derive a formula involving the Hermitian pairing of fusion trees. This formula verifies that non-semisimple quantum knot invariants can be explicitly encoded via the language of fusion trees in the NSS TQC mathematical architecture. |
| title | Fusion Trees and Homological Representations |
| topic | Geometric Topology Quantum Algebra 18M15, 18M20, 18M30, 57K16, 81R50 |
| url | https://arxiv.org/abs/2510.27218 |