The Teichmüller TQFT volume conjecture for twist knots

Fathi Ben Aribi; Eiichi Piguet-Nakazawa

doi:10.1016/j.crma.2019.02.004

Topology/Geometry

The Teichmüller TQFT volume conjecture for twist knots

Presented by: Claire Voisin

Fathi Ben Aribi ¹; Eiichi Piguet-Nakazawa ¹

¹ Université de Genève, Section de mathématiques, 2–4, rue du Lièvre, case postale 64, 1211 Genève 4, Switzerland

Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 299-305.

Abstract
Résumé

The Teichmüller TQFT, defined by Andersen and Kashaev, gives rise to a quantum invariant of triangulated hyperbolic knot complements; it has an associated volume conjecture, where the hyperbolic volume of the knot appears as a certain asymptotic coefficient.

In this note, we announce a proof of this volume conjecture for all twist knots up to 14 crossings; along the way we explicitly compute the partition function of the Teichmüller TQFT for the whole infinite family of twist knots.

Among other tools, we use an algorithm of Thurston to construct a convenient ideal triangulation of a twist knot complement, as well as the saddle point method for computing limits of complex integrals with parameters.

La TQFT de Teichmüller, définie par Andersen et Kashaev, produit un invariant quantique des complémentaires de nœuds hyperboliques triangulés ; elle a une conjecture du volume associée, où le volume hyperbolique du nœud apparaît comme un certain coefficient asymptotique.

Dans cette note, nous annonçons une preuve de cette conjecture du volume pour tous les nœuds twist de 14 croisements ou moins ; nous calculons au passage explicitement la TQFT pour l'intégralité de la famille infinie des nœuds twist.

Entre autres outils, nous utilisons un algorithme de Thurston pour construire une triangulation idéale pratique du complémentaire d'un nœud twist, ainsi que la méthode du point selle pour calculer des limites d'intégrales complexes paramétrées.

Received: 2018-10-05
Accepted: 2019-02-25
Published online: 2019-03-01

DOI: 10.1016/j.crma.2019.02.004

Author's affiliations:

Fathi Ben Aribi ¹; Eiichi Piguet-Nakazawa ¹

¹ Université de Genève, Section de mathématiques, 2–4, rue du Lièvre, case postale 64, 1211 Genève 4, Switzerland

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     author = {Fathi Ben Aribi and Eiichi Piguet-Nakazawa},
     title = {The {Teichm\"uller} {TQFT} volume conjecture for twist knots},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {299--305},
     publisher = {Elsevier},
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     number = {3},
     year = {2019},
     doi = {10.1016/j.crma.2019.02.004},
     language = {en},
}

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Fathi Ben Aribi; Eiichi Piguet-Nakazawa. The Teichmüller TQFT volume conjecture for twist knots. Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 299-305. doi : 10.1016/j.crma.2019.02.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.02.004/

References
Cited by

[1] J.E. Andersen; R. Kashaev A TQFT from quantum Teichmüller theory, Commun. Math. Phys., Volume 330 (2014) no. 3, pp. 887-934

[2] J.E. Andersen; J.K. Nissen Asymptotic aspects of the Teichmüller TQFT, Trav. Math., Volume 25 (2017), pp. 41-95

[3] F. Ben Aribi, E. Piguet-Nakazawa, The Teichmüller TQFT volume conjecture for twist knots, in preparation.

[4] R. Kashaev The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys., Volume 39 (1997) no. 3, pp. 269-275

[5] R. Kashaev, The Teichmüller TQFT, invited lecture at the ICM 2018 in Rio de Janeiro.

[6] H. Murakami An introduction to the volume conjecture, Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory, vol. 541, 2011, pp. 1-40

[7] H. Murakami; J. Murakami The colored Jones polynomials and the simplicial volume of a knot, Acta Math., Volume 186 (2001) no. 1, pp. 85-104

[8] H. Murakami; Y. Yokota Volume Conjecture for Knots, Springer, 2018

[9] E. Piguet-Nakazawa, The Teichmüller TQFT volume conjecture for knots in lens spaces, in preparation.

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