LMO Spectrum: Quantum Invariants Framework

• LMO Spectrum is a formalism that categorifies the classical LMO invariant by integrating factorization homology, homotopy E₃-algebras, and quantum surgery theory to analyze 3-manifold invariants.
• The framework leverages the homotopy E₃ structure of Jacobi diagram algebras, applying the tetrahedron principle and excision axioms to rigorously justify surgical decompositions in topology.
• Its computational strength is demonstrated by operator-valued refinements that distinguish subtle topological differences, such as in specific lens spaces, thereby advancing quantum topology.

The LMO Spectrum is a formalism introduced for the categorification of the Le-Murakami-Ohtsuki (LMO) invariant, integrating techniques from factorization homology, higher algebraic structures (specifically homotopy E₃-algebras), and quantum surgery theory. It supplies both an axiomatically grounded computational framework and a platform unifying geometric and algebraic approaches to 3-manifold invariants (Kuriya, 26 Aug 2025). The spectrum emerges in several contexts: as a quantum topological invariant, as an operator-valued refinement of classical numerical invariants, and as a manifestation of deep geometric principles underlying Jacobi diagram algebras and their surgery-theoretic calculus.

1. Algebraic Structure: Homotopy E₃-Algebra of Jacobi Diagrams

Central to the LMO Spectrum is the proof (Theorem A) that the Jacobi diagram algebra $\mathcal{A}_{\mathrm{Jac}}$ possesses a homotopy E₃-algebra structure. This result arises from interpreting the IHX relation in the algebra not merely as a combinatorial identity but as a direct algebraic avatar of the topology of the 3-dimensional tetrahedron—a principle termed the "Tetrahedron Principle." The E₃-algebra structure formalizes the existence of a compatible collection of $n$-ary bracket and multiplication operations, with coherences governed by the configuration spaces of points in $\mathbb{R}^3$.

The implication is that $\mathcal{A}_{\mathrm{Jac}}$ is not a mere commutative algebra but carries additional "braiding" and "homotopical" structure, reflecting the possibility of moving and intertwining diagram vertices in 3-space. This is leveraged to rigorously justify the "Principle of Decomposability"—the geometric axiom underlying Goussarov-Habiro clasper calculus—which ensures that surgical operations on 3-manifolds decompose consistently at the algebraic level.

2. Factorization Homology and Excision in 3-Manifold Topology

The computational foundation of the LMO Spectrum is built from first principles of factorization homology. For a closed oriented 3-manifold $M$, the factorization homology $\int_M \mathcal{A}_{\mathrm{Jac}}$ represents a canonical way of gluing local observables (constructed from Jacobi diagrams) throughout $M$. The excision axiom (Theorem B) provides a universal surgery formula, allowing the value of the invariant on a glued manifold to be computed in terms of its values on the pieces and the gluing data, independently of any conjectural models.

This formalism encodes how Jacobi diagrams propagate under surgery and provides a topological quantum field theory (TQFT)-style approach: closed diagrams correspond to scalar invariants, while open diagrams act as operators between labeled boundary objects. The homotopy E₃-algebra structure is essential for the compatibility of these operations under gluing.

3. The Tetrahedron Principle and IHX Relations

The IHX relation is interpreted as expressing the essential combinatorial property of the 3-simplex (tetrahedron). The analogy to associators in Drinfeld-Kontsevich theory is formal: Furusho's work (0903.4067) demonstrates that solutions of the pentagon equation (associated with the algebraic structure of associators) automatically satisfy the hexagon relation, encapsulating the higher associativity and commutativity needed for the diagram calculus. This result is transplanted into the Jacobi diagram setting: the geometric configuration space associated with the tetrahedron induces the algebraic relations essential for consistent surgery operations and the construction of invariants.

This connection bridges quantum group theory, the theory of associators, and finite-type invariants of 3-manifolds, situating the LMO Spectrum at an intersection point of algebraic and geometric topology.

4. The H₁-Decorated LMO Invariant and Operator-Valued Observables

As an applied outcome (Theorem C), the framework produces the "H₁-decorated LMO invariant," an operator-valued refinement of the classical LMO invariant. In accordance with TQFT principles, closed diagrams yield numerical invariants (the trace of the operator associated with the open diagram). The power of this refinement is demonstrated by its ability to distinguish pairs of lens spaces (specifically $L(156,5)$ and $L(156,29)$) that are indistinguishable by the standard LMO invariant—a limitation previously encountered in quantum topology (Kuriya, 26 Aug 2025).

The existence of such observable-valued invariants presages a systematic hierarchy of quantum invariants, leading to a richer “spectrum” that reflects finer topological information than scalar-valued invariants alone.

5. Consistency with Geometric Surgery Theory: Clasper Calculus

The factorization homology framework, grounded by the homotopy E₃-algebra structure, supplies a mathematical justification for the Goussarov-Habiro clasper surgery theory. The Principle of Decomposability, previously a postulate based on geometric intuition, is rigorously proven as a consequence of the algebraic structure and the excision property in factorization homology.

This not only guarantees consistency in geometric-topological surgery operations but also ensures that the process of generating 3-manifolds through surgery and their associated invariants is controlled by deep algebraic principles, in particular those surfacing through the configuration spaces of 3-manifolds.

6. Computational Utility and Impact in Quantum Topology

The axiomatically grounded LMO Spectrum framework gives rise to a universal surgery formula enabling explicit computation of higher-order invariants, operator-valued (observable) invariants, and their traces, all within a single coherent machinery. The approach is independent of conjectural constructs, ensuring reliability and universality. It opens new avenues for distinguishing subtle topological structures and encoding fine quantum invariants unattainable by scalar-valued constructs.

The unification of geometric intuition from clasper theory and algebraic rigor via higher algebra provides an extensible base for future work in both the mathematical and physical aspects of quantum topology.

7. Theoretical Implications and Future Directions

The formalism indicates that the categorification of finite-type invariants is well-governed by higher algebraic structures, with the E₃-algebra of Jacobi diagrams serving as a central organizing principle. This insight suggests further applications in categorified TQFTs, mapping class group actions, and possibly even in higher-dimensional manifold invariants.

The explicit demonstration that decorated invariants (operator-valued) refine previous quantum invariants points to a systematic methodology for producing ever finer stratifications (“spectra”) of topological invariants tied to algebraic and geometric structures of the underlying manifolds.

In summary, the LMO Spectrum formalism establishes a conceptual and computational framework for understanding and extending the LMO invariant, with categorical, algebraic, and geometric underpinnings that clarify and enhance the landscape of quantum invariants for 3-manifolds and beyond (Kuriya, 26 Aug 2025).