Fusion Trees and Homological Representations

Sung Kim

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arXiv · 2510.27218

Fusion Trees and Homological Representations

Sung Kim

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Abstract

We establish an identification between the spaces of $\alpha$-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.

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BibTeX RIS

Sung Kim. 2025-10-31. Fusion Trees and Homological Representations. https://arxiv.org/abs/2510.27218

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