The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra

Takahito Kuriya

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arXiv · 2508.18985

The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra

Takahito Kuriya

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Abstract

We define the LMO spectrum, a categorification of the Le-Murakami-Ohtsuki (LMO) invariant for 3-manifolds, using factorization homology. The theoretical foundation is our main algebraic result (Theorem A): the algebra of Jacobi diagrams, $\AJac$, possesses a homotopy $E_3$-algebra structure. This is a necessary condition for consistency within factorization homology, and the proof relies on the formality of the little 3-disks operad. A universal surgery formula is derived from the excision axiom (Theorem B), providing a computational basis independent of conjectural models. As an application (Theorem C), we construct an ``$H_1$-decorated LMO invariant'' that distinguishes the lens spaces $L(156, 5)$ and $L(156, 29)$, a pair that the classical LMO invariant fails to separate.

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BibTeX RIS

Takahito Kuriya. 2025-10-06. The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra. https://arxiv.org/abs/2508.18985

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