Relative Gang-Yonekura Formula

• The Relative Gang-Yonekura Formula is a framework that relates topological, geometric, and spectral invariants by transforming the 3D index in contexts such as Dehn filling.
• It employs methodologies like the pentagon identity and inductive gluing schemes to ensure accurate computation of relative indices for triangulated 3-manifolds.
• This formula bridges quantum topology and automorphic representation theory, offering a unified tool for analyzing manifold invariants and their asymptotic behaviors.

The Relative Gang-Yonekura Formula designates a family of formulas that relate topological, geometric, and spectral invariants in the context of relative structures—initially arising in trace formula theory and, most recently, in quantum topology through the paper of the 3D index for 3-manifolds under Dehn filling (Celoria et al., 11 Sep 2025). Its prototypes in automorphic representation theory (e.g., (Getz et al., 2014, Hahn, 2015)) establish correspondences between sums over geometric (orbital or period) data and spectral decompositions, extended to situations where symmetries or boundary conditions feature centrally. The relative version generalizes these ideas to cases with exposed boundaries, additional symmetry, or topological surgery. The most explicit realization is found in the transformation of the 3D index under Dehn filling, where the formula incorporates relative indices for ideal triangulations with boundary.

1. Conceptual Foundations

The classical Gang-Yonekura formula expresses transformation laws or trace identities for certain invariants (such as indices, partition functions, or automorphic periods) in the presence of a topological or spectral operation. In the relative setting, the “Relative Gang-Yonekura Formula” refers to:

• A formula that equates, under a relative operation (such as Dehn filling or restriction to a subgroup), modified invariants associated to a manifold or an automorphic quotient.
• The assignment of homological or spectral data to the relative pieces, an essential feature in quantum topology and representation theory.

This principle was established rigorously for the 3D index transformation under Dehn filling in (Celoria et al., 11 Sep 2025), bringing quantum topological and representation-theoretic methodologies into direct contact via the index’s generating function interpretation and period integral analogues.

2. Relative Index in Quantum Topology

The 3D index, initially introduced by Dimofte, Gaiotto, and Gukov, is a $q$-series encoding geometric and topological information about cusped 3-manifolds. The “relative” 3D index generalizes this by considering an ideal triangulation with exposed boundary, allowing for:

• Computation of the index through summation over solutions to Q-normal surface equations with boundary data fixed.
• Recovery of the global index through a gluing procedure that matches boundary assignments between triangulated pieces via omitted edge sets (corresponding to maximal trees in the 1-skeleton).

The gluing principle developed in (Celoria et al., 11 Sep 2025) establishes that the relative index behaves well under the concatenation of manifold pieces, providing the technical foundation for the Relative Gang-Yonekura Formula in this context.

3. The Formula for Dehn Filling

When performing Dehn filling (replacing a cusp by a layered solid torus along a slope $\alpha$ with dual curve $\alpha^*$), the Relative Gang-Yonekura Formula dictates how the index transforms:

$$\mathrm{GY}(\alpha; b) = \sum_k (-1)^k\, \left[q^{k/2} I_\mathrm{rel}(k\alpha; b) - I_\mathrm{rel}(k\alpha + 2\alpha^* ; b) \right]$$

• The summation index $k$ is restricted by congruence conditions from the specific boundary coefficients $b$.
• Terms only contribute when the intersection number $i(\alpha, \gamma)$ equals $0$ or $\pm2$.
• $I_\mathrm{rel}$ denotes the relative index computed from the Q-normal solution space, reflecting the topology and combinatorics of the triangulation.

This formula is established inductively for general layered solid tori by leveraging the pentagon identity—a three-term relation for the tetrahedral index—which underlies the validity of the formula under local moves (right or left insertions of tetrahedra).

4. Methods of Proof and Computational Implementation

A central aspect of the proof of the Relative Gang-Yonekura Formula is:

• The use of the pentagon identity to establish the behavior of the index under iterative construction of layered solid tori.
• An inductive scheme: If the formula holds for one configuration, pentagon moves provide its validity for any extended word in the letters $R$ and $L$ representing layered solid tori.
• Deployment of the Garoufalidis-Kashaev meromorphic extension for analytic control of the index.

Computationally, the relative index may be approached via two main strategies:

Method	Description	Tool Examples
Surface approach	Summation over Q-normal surfaces	Normaliz, Regina
Edge weights	Assignment via triangulation edge weights	SnapPy, Sage, Mathematica

Both methods admit certified computation up to a specified $q$-degree, enabling rigorous verification for the index transformation in a wide range of explicit cases.

5. Asymptotic and Homological Behavior

The Relative Gang-Yonekura Formula enables the paper of limiting behaviors of the 3D index under families of fillings:

• When the filling slope $\alpha_n = \alpha + n\beta$ varies, the limit depends on whether the sequence converges to a rational or irrational boundary slope.
• Specific examples (such as alternating torus knots and the figure-eight knot complement) illustrate the formula:
  • For $(p, q)$ torus knots, the index acts as a delta-function on boundary homology: $I^{(x, y)}(q) = \delta_0(x + p \cdot q \cdot y)$.
  • For longitudinal surgery on the figure-eight knot, the closed index is exactly $1$, matching known topological invariants.

Manifolds that are “toroidal” or closed can exhibit divergent constants, but non-constant coefficients stabilize in the $q$-series expansion, indicating robust topological encapsulation in the index.

6. Connections to Relative Trace Formulas

The mechanistic parallels between the formula for quantum indices and relative trace formulas in automorphic representation theory are notable:

• Both establish an isomorphism between geometric data (orbit integrals, boundary assignments) and spectral information (representational periods or index coefficients).
• Relative trace formulas (Getz et al., 2014, Hahn, 2015) utilize test functions, period integrals, and orbital classes to isolate spectral contributions associated with subgroups or twisted symmetries; the relative index gluing theorem and boundary data play analogous roles in quantum topology.
• The technical requirements (e.g., ellipticity, unimodularity, and closedness for relative trace formulas; admissible boundary assignments and congruence conditions for the index) ensure both convergence and meaningful spectral/geometric correspondence.

A plausible implication is that the algebraic and analytic machinery underpinning these trace formulas could eventually yield additional, deeper invariance or classification results for quantum topological objects.

7. Open Questions and Future Directions

Current research, as documented in (Celoria et al., 11 Sep 2025), points to several unresolved problems:

• The physical or direct topological interpretation of index coefficients remains elusive.
• Extension of the formula to multi-cusped or closed manifolds raises questions about the full range of possible index values.
• It is conjectured that constant values (0, 1, 2) classify closed indices in different topological settings.
• Understanding whether the index can distinguish between different types of closed 3-manifolds is a continuing area of investigation.

Moreover, the synthesis of relative trace formulas and topological index theories may motivate further developments in the classification and computation of invariants across disparate mathematical domains.

Summary Table: Key Elements of the Relative Gang-Yonekura Formula

Element	Description	Domain
Relative index	q-series encoding boundary-informed geometric data	Quantum topology
Gluing principle	Assembling global invariants from boundary pieces	Triangulated 3-manifolds
Period integrals	Integrals over subgroups capturing spectral data	Automorphic forms
Pentagon identity	Inductive relation for index stability	Layered solid tori
Homological constraints	Conditions on summation (intersection, congruences)	Surgery curves, triangulations

This framework situates the Relative Gang-Yonekura Formula at the intersection of quantum topology, automorphic representation theory, and spectral geometry, providing a unified language for the transformation and computation of deep invariants under relative or local operations.