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Bailey Pairs for the Tetrahedron Index

Ilmar Gahramanov, Sinan Ulaş Öztürk, Uveys Turhan

Abstract

In this work, we develop new Bailey pairs for the pentagon identity satisfied by the tetrahedron index, expressible in terms of $q$-series. Since the tetrahedron index underlies topological invariants of 3-manifolds and related knots, our construction may offer a new framework to deriving knot invariants through the Bailey chain.

Bailey Pairs for the Tetrahedron Index

Abstract

In this work, we develop new Bailey pairs for the pentagon identity satisfied by the tetrahedron index, expressible in terms of -series. Since the tetrahedron index underlies topological invariants of 3-manifolds and related knots, our construction may offer a new framework to deriving knot invariants through the Bailey chain.
Paper Structure (3 sections, 2 theorems, 15 equations)

This paper contains 3 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Bailey Pairs
  3. Conclusions

Key Result

Theorem 1.1

Let $m_i, e_i \in \mathbb{Z}$, then ${I_{\triangle}}(m,e)$ obeys the pentagon identity Dimofte:2011pyGaroufalidis2012The3I: with the balancing condition is

Theorems & Definitions (4)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.1
  • proof