Gang-Yonekura Formula in 3D Index Transformations

• Gang-Yonekura formula is a robust analytic framework that transforms the 3D index of cusped 3-manifolds under Dehn fillings using combinatorial geometry and q-series techniques.
• It expresses the filled manifold’s index as a sum over Q-normal surface contributions filtered by a kernel function, leveraging tetrahedral indices and Neumann–Zagier symplectic relations.
• The approach provides an inductive proof framework and explicit computational algorithms that validate asymptotic behaviors and ensure topological invariance across various filling regimes.

The Gang-Yonekura formula describes the transformation properties of the 3D index—a q-series invariant encoding topological and geometric data—under Dehn filling operations on cusped orientable 3-manifolds. Originally motivated by physics through the work of Dimofte, Gaiotto, and Gukov, the formula rigorously connects the index of a filled manifold to that of its parent cusped manifold, with crucial dependencies on both combinatorial and analytic structures arising from triangulations, normal surface theory, and q-hypergeometric functions.

1. Theoretical Foundations

The 3D index $I(M)$ for a cusped 3-manifold $M$ is structured as a generating function over (minimal) Q-normal surfaces, utilizing integer data attached to peripheral curves and solutions to the Q-matching equations. These conditions employ Neumann–Zagier symplectic relations, yielding key invariants: the formal Euler characteristic and the double curve count, which jointly govern the exponent of $q$ for each summand.

A “relative” version of the 3D index is defined for manifolds with exposed boundary. The relative index fixes peripheral boundary data (integer coefficients) and sums over Q-normal surfaces compatible with those constraints. This generalization facilitates an analytic approach to transformations under Dehn filling.

2. Statement and Structure of the Gang-Yonekura Formula

The Gang-Yonekura formula expresses the index of a Dehn-filled manifold (with the relevant cusp replaced by a layered solid torus) as a sum over selected contributions from the original cusped manifold:

• The kernel function acts as a sieve, isolating those contributions for which the intersection number with the filling slope equals $0$ or $\pm 2$, with the sign and $q$-exponent determined by the number of boundary components.
• The formula admits a "compact" representation using Q-normal surfaces and is verified to respect local moves ("R" and "L" layering in the context of layered solid tori) through explicit compatibilities.

3. Inductive Proof and Handling Degeneracies

A pivotal methodological advance is the development of a gluing principle for relative indices, allowing induction on the length of the word in R/L moves representing sequences of Dehn fillings. For each addition of a layer, the relative index is updated via a sum involving the tetrahedral index and a controlling $q$-exponential factor. The transformation is shown to be compatible by leveraging the pentagon identity and three-term relations for tetrahedral indices (notably in Proposition 8.12).

The induction's base case examines the degenerate layered solid torus associated with the Farey triple $(0,1,1)$. It is established (Theorem A.2) that the 3D index vanishes in this scenario, with careful handling necessary for degenerate cases via non-minimal triangulations to ensure rigor in the induction.

4. Analytic Identities and Hypergeometric Techniques

Central to the analysis is the generating function $\varphi_{(r)}(z,q)$, constructed as a shifted sum of tetrahedral indices (tracing a diagonal in $\mathbb{Z}^2$). Utilizing machinery from basic q-hypergeometric series, a critical identity (Theorem 8.21)—the "trigonometric identity" or q-Pythagorean theorem—demonstrates that a specified combination of such generating functions evaluates to a monomial. Further linear relations (Corollary 8.22) allow consistency checks and lead to closed-form summations, strengthening the theoretical framework.

5. Asymptotics under Large Dehn Fillings

Two distinct asymptotic regimes for the 3D index under Dehn filling are rigorously analyzed:

• Convergent Rational Slopes: For slopes tending toward a rational value in the Farey graph, the index limit corresponds to the difference of boundary indices (Theorem 9.1).
• Irrational Slopes: For fillings along continued fraction convergents to an irrational slope, the index converges to that of the unfilled manifold with trivial boundary conditions (Theorem 9.2).

This dichotomy extends understanding of the topological invariance and stability of the index, with implications for quantum topology.

6. Explicit Examples and Computational Validation

Explicit computations for alternating torus knots show the index is trivial (a delta-function on the boundary), providing verification of earlier conjectures. For more intricate cases, such as the Whitehead link and the “magic” manifold, the index values obtained via the formula match results from 1-efficient triangulations. Instances are observed where differing surgery descriptions for the same manifold yield identical 3D indices.

Two computational methodologies are detailed:

• The surface approach enumerates minimal Q-normal surfaces using combinatorial geometry software such as Normaliz, Regina, and SnapPy.
• The edge weights approach computes multi-sums over integer edge weight assignments, accommodating half-integer corrections from peripheral homology.

Key algorithmic techniques include Hermite decomposition, unimodular triangulation of polyhedral cones, and efficient enumeration routines, enabling accurate calculation of the index to any prescribed degree $D$. Code is provided—primarily in Mathematica, complemented by Sage, Normaliz, Regina, and SnapPy—for certified index computations.

7. Open Questions and Directions

The work substantiates the Gang-Yonekura formula for closed 3-manifolds derived via Dehn filling, addresses challenges from degenerate tori and convergence, and confirms the index’s topological invariance through extensive q-series and hypergeometric analysis. Potential avenues for further research include:

• Extensions to manifolds with multiple cusps and the resulting interplay of quantum invariants.
• Deepened topological interpretations and connections to non-hyperbolic or toroidal manifolds.
• Investigation of invariance and extension properties across varied triangulation schemes.

A plausible implication is that the established formalism could be generalized to a broader class of quantum invariants or applied to questions in quantum topology beyond the current contexts.

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In sum, the Gang-Yonekura formula establishes a precise analytic and combinatorial framework for tracking the transformation of quantum invariants under Dehn filling, offering robust computational and theoretical tools that reinforce its centrality in the paper of 3-manifold topology.