3D Index: Topology, Quantum Invariants & Data

• 3D Index is a set of constructs that link q-series invariants in 3-manifolds with computational methods for 3D object retrieval.
• The index is computed from ideal triangulations using tetrahedron indices, and is validated through normal surface theory and Pachner moves.
• Techniques such as holonomic q-difference equations and PCA-based feature extraction bridge quantum topology with modern data science applications.

The term 3D Index designates several fundamentally distinct, research-level constructs arising in topology, geometry, mathematical physics, and data science. Most prominently, it refers to a q-series invariant for 3-manifolds with torus boundary—originally appearing in the context of quantum topology, normal surface theory, and supersymmetric quantum field theory—and it encompasses methods for computational retrieval in 3D object databases via projections to lower-dimensional feature spaces. The detailed structure, analytical foundations, and applications of the 3D index span rigorous axiomatic definitions, computational frameworks for manifold invariants, and connections to dualities and partition functions in supersymmetric gauge theory.

1. Definition and Core Constructions

Quantum Topological 3D Index

The 3D index, as introduced by Dimofte, Gaiotto, and Gukov, associates to an ideal triangulation of a compact oriented 3-manifold ($M$) with torus boundary a collection of formal power series in $q^{1/2}$, with integer coefficients. For each choice of peripheral data, the index is constructed by associating to each tetrahedron a tetrahedron index (a specific $q$-hypergeometric series) and summing over a lattice of normal surface weights compatible with the triangulation constraints. The construction relies on the matching equations of normal surface theory, with careful quotienting to remove trivial ("tetrahedral") solutions and peripheral contributions (Garoufalidis, 2012, Garoufalidis et al., 2016).

For a fixed triangulation $T$ with $n$ tetrahedra and $r$ torus boundary components, the 3D index is built as:

$$\mathcal{I}_T(m, e)(q) = \sum_{k} q^{\ast} \prod_{j=1}^n J(a_j, b_j, c_j), \quad J(a, b, c) \in \mathbb{Z}[[q^{1/2}]]$$

where the $J(a, b, c)$ are cyclically symmetric tetrahedron indices, $k$ runs over an appropriate quotient lattice, and $(m, e)$ parameterize the boundary charges (e.g., meridian and longitude).

Data-Driven 3D Indexing (Computer Vision)

In 3D model retrieval, the “3D index” is a feature descriptor extracted from a multi-view representation of a 3D shape as a collection of 2D depth images rendered from points on a view sphere. These depth images are processed—using Principal Component Analysis (PCA), Independent Component Analysis (ICA), or Nonnegative Matrix Factorization (NMF)—to produce a compact, invariant feature embedding for retrieval (Dutagaci et al., 2011).

2. Properties and Mathematical Structures

Holonomicity and Recursion

The set of tetrahedron indices forms a q-holonomic module (i.e., solutions to finite-order $q$-difference equations), subject to recursive relations and a parity condition: $$q^{e/2}f(m+1, e) + q^{-m/2} f(m, e+1) - f(m, e) = 0$$ Fulfilling triality symmetries and possessing closed formulas in terms of $q$-hypergeometric summations, these play a central role in ensuring that the total 3D index inherits a holonomic structure (Garoufalidis, 2012).

Invariance under Moves

Topological invariance of the 3D index is guaranteed (for 1-efficient triangulations) under Pachner 3–2 moves, while 2–3 moves require additional geometric constraints (such as a "special" angle structure). This underpins its status as a topological invariant of the underlying 3-manifold, rather than a combinatorial artifact of its triangulation (Garoufalidis, 2012, Garoufalidis et al., 2016).

Gluing and Relative Indices

For the paper of Dehn fillings and cutting-gluing operations, a relative 3D index is defined on ideal triangulations with exposed boundary, enabling the rigorous assembly of the index for a closed manifold from those of its pieces (Celoria et al., 11 Sep 2025). This gluing principle extends the applicability of the index to contexts beyond those originally accessible to the Dimofte–Gaiotto–Gukov construction.

3. Transformations under Dehn Filling: The Gang–Yonekura Formula

A central achievement is the rigorous proof of the Gang–Yonekura formula for Dehn surgery. Given a filling slope $\alpha$ and its dual $\beta$, the 3D index $\mathcal{I}_{\text{filled}}$ of the filled manifold is given in terms of the relative index $\mathcal{I}^{\text{rel}}_C$ on the standard cusp: $$\mathcal{I}_{\text{filled}}(\mathbf{b}) = \sum_k (-1)^k \left[ q^{k/2} \mathcal{I}_C^{\text{rel}}(k\alpha; \mathbf{b}) - \mathcal{I}_C^{\text{rel}}(k\alpha+2\beta; \mathbf{b}) \right]$$ with $k$ running over integers subject to parity restrictions (Celoria et al., 11 Sep 2025).

The proof is based on a gluing theorem for relative indices (matching the combinatorics of normal Q-surfaces), identities for generating functions of tetrahedron indices involving basic $_3\phi_3$ hypergeometric series, and analytic continuation via a meromorphic extension of the index (Garoufalidis et al., 2017). This formula allows for the computation of the 3D index of closed manifolds, even in the absence of a direct intrinsic definition.

4. Examples, Computational Methods, and Topological Invariance

Certified computation of the 3D index is achieved by combining:

• Extraction of triangulation and gluing data using SnapPy,
• Hilbert basis computations and cone decomposition for normal surfaces via Normaliz and Regina,
• Surface-based and edge-weight-based enumeration of Q-normal classes, with degree bounds for efficiency.

Benchmarks verify the invariance of the index across distinct triangulations (supporting topological invariance), and explicit calculations confirm expected behaviors for manifolds such as torus knots (index "delta-functions" enforcing linear relations), Whitehead link complements, and filled figure-eight knot complements (index identically $1$ for certain surgeries) (Celoria et al., 11 Sep 2025).

Large-filling and asymptotic behaviors corroborate the interpretation of the 3D index as a quantum generating function for normal surface classes, with limiting behaviors reflecting classical topological features and the suppression of peripheral contributions.

5. Structural and Analytical Implications

Quantum Topological Interpretations

The 3D index, in its sum-over-normal-surfaces form, bridges quantum topology, normal surface theory, and quantum field theory. The strict constraints for 1-efficient triangulations ensure convergence and alignment with geometric (e.g., hyperbolic) structures.

The meromorphic extension developed in (Garoufalidis et al., 2017) provides analytic continuation and new proofs of invariance, demonstrating that the 3D index fits into the landscape of quantum non-semisimple invariants, state-integrals, and modularity properties relevant in quantum topology and field theory.

Connections to Physical Theories

The 3D index emerges in the 3d–3d correspondence, relating q-series invariants for 3-manifolds to partition functions of supersymmetric 3d field theories and Chern–Simons theories on mapping tori (Hwang et al., 2012, Gang et al., 2013, Duan et al., 2022). The Gang–Yonekura formula, in particular, fulfills a supersymmetry prediction about the behavior of BPS state counts under surgery.

Bilinear factorization into holomorphic blocks, difference equations mirroring quantized character varieties, and interplay with quantum modular forms link the 3D index to deep arithmetic and geometric structures.

6. Extensions and Open Questions

Several open questions and avenues for further research are highlighted:

• Defining the 3D index intrinsically for closed 3-manifolds (without reliance on filling constructions),
• Interpreting index coefficients with respect to geometric features (e.g., detecting reducibility or the presence of essential tori),
• Understanding its categorification and connections to quantum modularity,
• Systematizing the relative index and its gluing in higher complexity cases (multiple cusp fillings),
• Establishing direct relationships with quantum invariants arising from field-theoretic constructions and modular tensor category data.

Overall, the 3D index—across its incarnations—serves as a bridge between quantum invariants, classical topology, and modern field theory, providing computable, topologically robust invariants with rich algebraic and physical content.