2025/09/11 by Daniele Celoria, Craig D. Hodgson, Celoria, Daniele +4 · 1 voice · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Clustering Algorithms Research #Advanced Database Systems and Queries #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Number Theory (math.NT) #Topological and Geometric Data Analysis #hep-th #math.GT #math.NT
paper · pdf · doi:10.48550/arxiv.2509.09886
openalex publication_date 2025/09/11 · arxiv published 2025/09/11 · openalex created_date 2025/10/10 · arxiv updated 2025/12/19 · openalex updated_date 2026/07/28
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.