Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the tetrahedral index and the Hahn-Exton q-Bessel function

Daniele Celoria

TL;DR

This note establishes a precise link between the tetrahedral index $I_\Delta(m,e)$ and the Hahn Exton $q$-Bessel function $J_\nu(z;q)$ via $I_\Delta(m,e)=J_e(q^{-m/2};q)$, enabling the transfer of $q$-hypergeometric identities across the two frameworks. It shows that fundamental relations satisfied by $J_\nu(z;q)$, such as three-term recurrences, quadratic and pentagon identities, yield nontrivial identities for $I_\Delta$, and conversely that $I_\Delta$ identities can be translated into new $J_\nu$ consequences. The correspondence is then used to reinterpret the topological 3D index under Dehn fillings, revisiting Gang and Yonekura's Dehn-filling formula and framing it in a number-theoretic context, with potential implications for convergence and boundary data. Finally, the note discusses conjectural relations in the closed case and open questions about whether purely $q$-hypergeometric methods can detect exceptional surgeries, inviting collaboration between low-dimensional topology and $q$-series communities.

Abstract

The purpose of this short note is twofold: First to elucidate some connections between the ``building block'' of Dimofte--Gaiotto--Gukov's $3$D index, known as the tetrahedral index $I_Δ(m,e)$, and Hahn--Exton's $q$-analogue of the Bessel function $J_ν(z;q)$. The correspondence between $I_Δ$ and $J_ν$ will allow us to translate useful relations from one setting to the other. Second, we want to introduce to the $q$-hypergeometric community some possibly new techniques, theory and conjectures arising from applications of physical mathematics to geometric topology.

A note on the tetrahedral index and the Hahn-Exton q-Bessel function

TL;DR

This note establishes a precise link between the tetrahedral index and the Hahn Exton -Bessel function via , enabling the transfer of -hypergeometric identities across the two frameworks. It shows that fundamental relations satisfied by , such as three-term recurrences, quadratic and pentagon identities, yield nontrivial identities for , and conversely that identities can be translated into new consequences. The correspondence is then used to reinterpret the topological 3D index under Dehn fillings, revisiting Gang and Yonekura's Dehn-filling formula and framing it in a number-theoretic context, with potential implications for convergence and boundary data. Finally, the note discusses conjectural relations in the closed case and open questions about whether purely -hypergeometric methods can detect exceptional surgeries, inviting collaboration between low-dimensional topology and -series communities.

Abstract

The purpose of this short note is twofold: First to elucidate some connections between the ``building block'' of Dimofte--Gaiotto--Gukov's D index, known as the tetrahedral index , and Hahn--Exton's -analogue of the Bessel function . The correspondence between and will allow us to translate useful relations from one setting to the other. Second, we want to introduce to the -hypergeometric community some possibly new techniques, theory and conjectures arising from applications of physical mathematics to geometric topology.

Paper Structure

This paper contains 9 sections, 1 theorem, 44 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Tetrahedral index and the Hahn-Exton function
  4. The tetrahedral index
  5. The Hahn-Exton function
  6. From $J_\nu$ to $I_{\Delta}$
  7. From $I_{\Delta}$ to $J_\nu$
  8. $3$D index and Dehn fillings
  9. Conjectural relations in the closed case

Key Result

Proposition 2.1

Let $m,e \in \mathbb{Z}$; then

Figures (2)

  • Figure 1: The complement in $S^3$ of the figure-eight knot $4_1$ shown on the left, can be obtained by gluing two ideal tetrahedra, using the identifications displayed on the edges (cf.Thu78notes).
  • Figure 2: The complement of the knot $8_{16}$ (shown on the left) can be alternatively obtained e.g. by performing $(-1)$-framed Dehn surgery on the blue component of the link shown on the right. (Figure created using KLO KLO)

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Remark 2.3
  • Remark 3.1
  • Example 3.2
  • Example 4.1