The 3D index and Dehn filling

TL;DR

The paper provides a rigorous proof of the Gang–Yonekura formula describing how the 3D index transforms under Dehn filling of a cusp, framed via a new relative index for manifolds with exposed boundary. It develops a gluing principle for relative indices, proves an inductive framework using layered solid tori, and leverages Garoufalidis–Kashaev’s meromorphic extension with q-hypergeometric identities to establish the base case and inductive step. The work also introduces collar and 1-efficiency concepts to control degeneracies, analyzes large-filling asymptotics, and delivers certified computational tools that support index invariance and extendability to closed manifolds. Through extensive examples and computational validation, the authors demonstrate the index’s stability under surgeries and provide groundwork for a mathematically well-founded extension to closed 3-manifolds, with significant implications for quantum topology and computational topology.

Abstract

We provide a rigorous proof of the Gang-Yonekura formula describing the transformation of the 3D index under Dehn filling a cusp in an orientable 3-manifold. The 3D index, originally introduced by Dimofte, Gaiotto and Gukov, is a physically inspired q-series that encodes deep topological and geometric information about cusped 3-manifolds. Building on the interpretation of the 3D index as a generating function over Q-normal surfaces, we introduce a relative version of the index for ideal triangulations with exposed boundary. This notion allows us to formulate a relative Gang-Yonekura formula, which we prove by developing a gluing principle for relative indices and establishing an inductive framework in the case of layered solid tori. Our approach makes use of Garoufalidis-Kashaev's meromorphic extension of the index, along with new identities involving q-hypergeometric functions. As an application, we study the limiting behaviour of the index for large fillings. We also develop code to perform certified computations of the index, guaranteeing correctness up to a specified accuracy. Our extensive computations support the topological invariance of the 3D index and suggest a well-defined extension to closed manifolds.

Figures (15)

• Figure 1: The three normal quad types in a tetrahedron: separating vertices 01/23, 02/13 and 03/12 respectively.
• Figure 2: Gluing instructions for the two tetrahedra forming a standard cusp. The cusp vertex is in black, and the punctured torus forming the boundary of its neighbourhood is shaded.
• Figure 3: The labelling conventions used for $\mathcal{C}$. The unique internal edge is $e_0$, while $\lambda$ and $\mu$ are chosen as the longitude and meridian for the punctured torus boundary.
• Figure 4: There are two ways of completing a boundary curve to a spun annulus in the standard cusp. The two possibilities spin in opposite directions, and are related by an involution.
• Figure 5: The Farey ideal tessellation of the hyperbolic plane. Vertices are in bijection with reduced fractions in $\mathbb{Q} \cup \infty$. The dual tree is drawn with red/blue corresponding to left/right moves.

Theorems & Definitions (96)

• Theorem
• Proposition 2.1: howie2020polynomials
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4
• Theorem 3.1
• Definition 3.2
• Remark 3.3
• proof