Fusion Trees and Homological Representations

TL;DR

This paper links fusion trees in non-semisimple topological quantum computation with homological braid group representations, providing new proofs and formulas for quantum invariants using graphical calculus.

Contribution

It establishes a novel correspondence between fusion trees and homological braid group representations, enabling explicit encoding of quantum knot invariants.

Findings

Identifies fusion trees with Lawrence homological representations at roots of unity.

Provides a new proof of Ito's colored Alexander invariant formula.

Derives a fusion tree-based formula for quantum knot invariants.

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.