The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra (2508.18985v1)

Abstract: This paper introduces a framework for the categorification of the Le-Murakami-Ohtsuki (LMO) invariant of 3-manifolds, defining a new invariant, the LMO Spectrum, via factorization homology. The theoretical foundation of this framework is the main algebraic result of this paper (Theorem A): a proof that the algebra of Jacobi diagrams, A_Jac, possesses the structure of a homotopy E3-algebra. This algebraic structure is shown to be a consequence of 3-dimensional geometry, originating from a "Tetrahedron Principle" where the fundamental IHX relation is interpreted as an algebraic manifestation of the tetrahedron's combinatorial properties. This geometric perspective is used to provide a rigorous proof of the "Principle of Decomposability" that underpins the Goussarov-Habiro clasper calculus, thereby supplying a mathematical justification for the consistency of this geometric surgery theory. To establish a computational basis, the framework is built from the first principles of factorization homology, from which a universal surgery formula is derived via the excision axiom (Theorem B) independently of any conjectural models. The utility of this axiomatically-grounded theory is demonstrated by constructing a new observable, the "H_1-decorated LMO invariant." This invariant is defined in accordance with the principles of TQFT, where the evaluation of a closed diagram (a numerical invariant) is the trace of the corresponding open diagram (an operator). As the main applied result of this paper (Theorem C), it is proven that this invariant distinguishes the lens spaces L(156, 5) and L(156, 29), a pair known to be indistinguishable by the classical LMO invariant. This result validates the proposed framework and establishes the LMO Spectrum as a new tool in quantum topology that unifies the geometric intuition of clasper theory with the algebraic rigor of modern homotopy theory.