H₁-Decorated LMO Invariant in Quantum 3-Manifolds

• The H₁-decorated LMO invariant is a refined quantum invariant that normalizes torsion contributions by incorporating first homology data via a categorified operator-trace formalism.
• It constructs a diagrammatic algebra of labeled Jacobi diagrams that remains stable under band sum moves, ensuring robust functorial extensions and torsion correction.
• By enhancing distinguishing power among 3-manifolds, notably lens spaces, it paves the way for chromatic extensions and deeper insights into quantum topology.

The H₁-decorated LMO invariant refines the classical Le–Murakami–Ohtsuki invariant by systematically encoding the first homology group’s data within the algebraic and topological framework of quantum invariants for 3-manifolds. This construction, informed by the axioms and methodologies of factorization homology, replaces the extraction of a single universal perturbative invariant with a categorified operator formalism designed to resolve torsion phenomena and enhance distinguishing power among 3-manifolds, notably among lens spaces.

1. Universal Perturbative Framework and Torsion Correction

The LMO invariant ((M)), originally defined through surgery formulas and formal Gaussian integrals over trivalent Jacobi diagrams, is universal among finite-type (perturbative) invariants for rational homology 3-spheres. The central universality claim, established via explicit identities, is

$$((M)) = |H_1(M;\mathbb{Z})|^{\phi_+} \, \tau^{\mathfrak{g}}(M),$$

where $$\phi_+ = \frac{\dim \mathfrak{g} - \mathrm{rank} \mathfrak{g}}{2}$$ and $\tau^{\mathfrak{g}}(M)$ denotes the perturbative invariant associated to a simple Lie algebra $\mathfrak{g}$ (Kuriya et al., 2010). The torsion correction factor $|H_1(M;\mathbb{Z})|^{\phi_+}$ embodies the contribution of the first homology’s order and is factored out to produce the H₁-decorated LMO invariant: $$\tau^{\mathfrak{g}}(M) = \frac{((M))}{|H_1(M;\mathbb{Z})|^{\phi_+}}.$$ The decorated invariant thus normalizes the “quantum contribution” so that for integral homology spheres ($|H_1(M;\mathbb{Z})|=1$) the torsion factor is trivial and the two invariants coincide.

2. Diagrammatic Algebra, Functorial Extensions, and Band Sum Moves

The invariant is constructed in the group-like completed algebra of Jacobi diagrams, typically denoted $\mathcal{A}(H_1)$, whose diagrams now carry extra labels (beads, legs, or markings) recording the H₁-information. In the refined construction, the output of the invariant is an element of $\mathcal{A}(H_1)$, often written as an exponential series: $$\widetilde{Z}_f(M) = \exp \left( \sum_{\Gamma \in \mathcal{A}_{H_1}} c_\Gamma \cdot \Gamma \right)$$ where the coefficients $c_\Gamma$ depend on the chosen decoration function $f$ and encapsulate the behavior under topological operations (Gauthier, 2010, Gauthier, 2010). Under band sum moves—a key local surgery operation—the invariant transforms via

$$\widetilde{Z}_f(M') = \widetilde{Z}_f(M) \cdot \exp(\Delta(b)),$$

where $\Delta(b)$ encodes corrections tracking the interaction of the band with H₁ labels. In cases of homologically trivial moves, the correction is controlled or vanishes, reflecting the invariant’s stability under cobordism maneuvers.

3. Factorization Homology, E₃-Algebra, and Operator Formalism

Recent advances (Kuriya, 26 Aug 2025) establish a rigorous framework via factorization homology, categorifying the LMO invariant to a spectrum-level invariant with a homotopy E₃-algebra structure on the Jacobi diagram algebra. The LMO Spectrum is defined as

$$\mathrm{LMO}_{\mathrm{spec}}(M) := \int_M \mathcal{A}$$

and the H₁-decorated invariant becomes an observable adhering to TQFT principles: closed diagrams yield numerical invariants by tracing open diagram-induced operators on the state space $\mathcal{V} = \mathbb{C}[G]$, with $G = \mathrm{Tors}(H_1(M;\mathbb{Z}))$. For example, the $\theta$-graph evaluation is

$$\mathrm{ev}(\theta) = \frac{1}{|G|^3} \sum_{g_1 + g_2 + g_3 = 0} W(g_1, g_2, g_3)^2 \cdot \prod_{j=1}^3 \exp(2\pi i \cdot q_M(g_j)),$$

where $q_M$ is the linking quadratic form and $W$ is a universal vertex factor determined by algebraic consistency (IHX relation as a “Tetrahedron Principle”).

The underlying algebraic consistency and computability is guaranteed by rigorous verification of the E₃-algebra structure using modular operad arguments and homotopy transfer theorems, affirming the Principle of Decomposability essential for the Goussarov-Habiro clasper calculus.

4. Splitting, Functoriality, and Milnor Invariant Detection

In cobordism categories, the functorial extension (“LMO functor”) translates bottom-top tangle presentations into decorated Jacobi diagrams, respecting the classification of boundary components and genus (encoded via a free magma construction) (Nozaki, 2015). The decomposition into strut and Y-parts (linking matrix and higher-order terms) yields:

• $Z^s$: reduction to classical linking numbers, encoding H₁-data
• $Z^Y$: encodes finite-type and Milnor invariants (tree-level information of string links)

The functorial LMO invariant is universal among rational-valued finite-type invariants, with explicit surgery and clasper calculus establishing isomorphisms between graded diagram spaces and equivalence classes of cobordisms.

5. Spectral Distinction and Lens Space Phenomena

A principal application of the H₁-decorated invariant is its increased distinguishing power on lens spaces and rational homology spheres. Classical LMO invariants, due to their dependence on Dedekind sums, fail to separate certain lens spaces, notably $L(156,5)$ and $L(156,29)$. The H₁-decorated LMO invariant, constructed via operator-trace formalism and encoding explicit linking form data, successfully distinguishes these spaces (Kuriya, 26 Aug 2025). This sensitivity to torsion and subtle number-theoretic phenomena, inaccessible to unadorned quantum invariants, validates the new framework.

6. Algebraic and Topological Foundations

The consistency of the decorated invariant’s surgery theory relies on algebraic interpretations of topological moves, with the IHX relation grounded in the combinatorial structure of a tetrahedron (the “Tetrahedron Principle”). The decomposition of Jacobi diagrams into fundamental relations ensures that higher-order surgery operations (clasps, bands, etc.) have algebraically justified decompositions, adhering to the principle that all topological manipulations can be resolved within the diagrammatic E₃-algebra.

The connection to Massey products and Johnson-type homomorphisms suggests deeper links between the decorated invariant and the algebraic topology of 3-manifolds, possibly providing completeness for the classification of integral homology spheres.

7. Future Directions: Chromatic Extensions and Physical Dualities

Emerging research aims to extend the invariant’s sensitivity via chromatic lifts—replacing chain complexes with spectra, leading to the “Chromatic LMO Spectrum”: $$\mathrm{LMO}_{\mathrm{spec}_n}(M) := \int_M (\mathcal{A}_{\mathrm{spec}} \wedge \mathbb{E}_n)$$ where $\mathbb{E}_n$ is a Morava E-theory spectrum. This direction connects to stable homotopy theory and proposes bridges to string-theoretic phenomena (Gopakumar–Vafa invariants) and higher-order torsion invariants. The spectral approach offers a pathway to categorifying quantum invariants and linking classical invariants (e.g., Casson, Rohlin) to the chromatic perspective in algebraic topology.

Summary Table: H₁-Decorated LMO Invariant Features

Aspect	Classical LMO	H₁-decorated LMO
Torsion Sensitivity	Factor via $|H_1|^{\phi_+}$	Explicit, operator-trace formalism
Distinction Among Lens Spaces	Fails on $L(156,5)$ vs $L(156,29)$	Succeeds; detects number-theoretic differences
Algebraic Framework	Jacobi diagrams (E₃ structure conjectured)	Jacobi diagrams (E₃ structure proven)
Functoriality	Surgery formula, finite-type	Factorization homology, TQFT axioms
Topological Moves	Controlled under band sum, splittings	Decorated moves coded in operator algebra
Future Extensions	Perturbative invariants	Chromatic spectra, Massey products, string dualities

The H₁-decorated LMO invariant constitutes a rigorous, categorified enhancement of quantum topological invariants, incorporating first homology data to reveal torsion-dependent structure in 3-manifold topology, grounded in the homotopy-theoretic properties of Jacobi diagram algebras and factorization homology. The framework not only resolves specific classification problems (e.g., lens spaces) but also sets the stage for chromatic, Massey, and physical duality explorations in modern topology.