From 3-dimensional skein theory to functions near $\mathbb{Q}$
Garoufalidis, Stavros1; Lê, Thang T. Q.2
1International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen (China)
2School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA,
Annales de l'Institut Fourier, Online first, 18 p.

Motivated by the Quantum Modularity Conjecture and its arithmetic aspects related to the Habiro ring of a number field, we define a map from the Kauffman bracket skein module of an integer homology 3-sphere to the Habiro ring, and use Witten’s conjecture (now a theorem) to show that the image is an effectively computable module of finite rank that can be used to phrase the quantum modularity conjecture.

Motivé par la conjecture de modularité quantique et par ses aspects arithmétiques liés à l’anneau d’Habiro d’un corps de nombre, nous définissons une application du module d’écheveau d’une $3$-sphère d’homologie vers l’anneau d’Habiro. Nous utilisons la conjecture de Witten, qui est désormais prouvée, pour montrer que l’image est un module de rang fini explicitement calculable. Cette construction est utilisée pour reformuler la conjecture de modularité quantique.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3740
Classification: 57K10, 57K31
Keywords: knots, links, 3-manifolds, Jones polynomial, Kauffman bracket, skein theory, Witten’s Conjecture, Habiro ring, Quantum Modularity Conjecture, Volume Conjecture, functions “near” $\mathbb{Q}$, character varieties
Mots-clés : nœuds, liens, variétés de dimension 3, polynôme de Jones, polynômes crochet de Kauffman, théorie des écheveaux, conjecture de Witten, anneau d’Habiro, conjecture quantique de modularité, conjecture du volume, fonctions « près » de $\mathbb{Q}$, variétés de caractères

Garoufalidis, Stavros  1 ; Lê, Thang T. Q.  2

1 International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen (China)
2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA, 
Garoufalidis, Stavros; Lê, Thang T. Q. From 3-dimensional skein theory to functions near $\mathbb{Q}$. Annales de l'Institut Fourier, Online first, 18 p.
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     author = {Garoufalidis, Stavros and L\^e, Thang T. Q.},
     title = {From 3-dimensional skein theory to functions near $\mathbb{Q}$},
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