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Thesis Reference Thesis Teichmüller TQFT Calculations for Infinite Families of Knots PIGUET, Eiichi Abstract La théorie topologique des champs quantiques (TQFT) de Teichmüller a été construite en 2011 par Andersen et Kashaev. Il s'agit d'un invariant de complémentaires de nœuds triangulés et aussi d'un invariant de nœuds qui est une analogie en dimension infinie de l'invariant de Kashaev. Elle a une conjecture du volume associée, qui donne un lien entre les deux invariants ci-dessus et qui dit en outre que le volume du nœud apparaît comme un certain coefficient asymptotique. Dans cette thèse, nous allons dans un premier temps calculer la fonction de partition de la TQFT de Teichmüller pour les nœuds twist et nous prouvons la conjecture du volume. Ensuite, nous allons présenter des résultats similaires, mais pour une famille de nœuds fibrés dans des espaces lenticulaires. Enfin, la dernière partie est consacrée à l’étude des caractères dans la théorie de Teichmüller quantique. Reference PIGUET, Eiichi. Teichmüller TQFT Calculations for Infinite Families of Knots. Thèse de doctorat : Univ. Genève, 2021, no. Sc. 5554 DOI : 10.13097/archive-ouverte/unige:152763 URN : urn:nbn:ch:unige-1527637 Available at: http://archive-ouverte.unige.ch/unige:152763 Disclaimer: layout of this document may differ from the published version. 1 / 1 UNIVERSITE´ DE GENEVE` FACULTE´ DES SCIENCES Section de Math´ematiques Professeur Rinat Kashaev Teichm¨ullerTQFT Calculations for Infinite Families of Knots THESE` Pr´esent´ee`ala Facult´edes Sciences de l'Universit´ede Gen`eve pour obtenir le grade de Docteur `esSciences, mention Math´ematiques par Eiichi PIGUET de Gen`eve (GE) Th`eseNo 5554 GENEVE` Atelier d'impression ReproMail 2021 アイデアの秘訣は執念である。 湯川秀樹 Abstract This thesis contains three parts with new results. The first one deals with calculations of partition functions of the Teichm¨ullerTopological Quantum Field Theory (TQFT) for twist knots, and the second one for a family of fibered knots in lens spaces. The last part is devoted to the study of characters in quantum Teichm¨ullertheory. The Teichm¨ullerTQFT was constructed in 2011 by Andersen and Kashaev. It is an invariant of triangulated knot complements and also a knot invariant which is an infinite- dimensional analogue of the Kashaev invariant. It has an associated volume conjecture, which gives a link between the two invariants above and furthermore says that the volume of the knot appears as a certain asymptotic coefficient. In this thesis, we will at first build a new infinite family of ideal triangulations and H- triangulations for the family of hyperbolic twist knots. These triangulations give a new upper bound for the Matveev complexity of twist knot complements. We prove that the above ideal triangulations are geometric using the technique of Futer{Gu´eritaud,namely the study of the volume functional on the polyhedron of angle structures. We then use these triangulations to compute explicitly the partition function of the Teichm¨ullerTQFT and prove the associated volume conjecture for all hyperbolic twist knots, using the saddle point method. In a second step, we present similar calculations and results as for the twist knots, but this time for an infinite family of hyperbolic fibered knots in the lens spaces L(n; 1) with n > 1. For the ideal triangulation, we use the monodromy triangulation of Floyd and Hatcher, which is geometric. For H-triangulations, we introduce a method called T-surgery which allows to construct H-triangulations from special ideal triangulations. Quantum Teichm¨ullertheory made it possible to construct unitary projective representa- tions of mapping class groups of punctured surfaces on infinite-dimensional Hilbert spaces. It is expected that the trace of the associated quantum operators gives rise to invariants of mapping tori. The trace of a unitary operator is not always defined. However, we will see that one can give an interpretation of this trace, in the case of a pseudo-Anosov monodromy on the once-punctured torus, using the Teichm¨uller TQFT with some ideal triangulation of the mapping torus provided with an \almost" complete structure. i Resum´ e´ Cette th`esecomporte trois parties avec de nouveaux r´esultats.La premi`ereporte sur des calculs de fonctions de partition de la th´eorie topologique des champs quantiques (TQFT) de Teichm¨ullerpour les nœuds twist, puis la deuxi`emepour une famille de nœuds fibr´es dans des espaces lenticulaires. La derni`erepartie est consacr´ee`al'´etudedes caract`eres dans la th´eoriede Teichm¨ullerquantique. La TQFT de Teichm¨ullera ´et´econstruite en 2011 par Andersen et Kashaev. Il s'agit d'un invariant de compl´ementaires de nœuds triangul´eset aussi d'un invariant de nœuds qui est une analogie en dimension infinie de l'invariant de Kashaev. Elle a une conjecture du volume associ´ee,qui donne un lien entre les deux invariants ci-dessus et qui dit en outre que le volume du nœud appara^ıtcomme un certain coefficient asymptotique. Dans cette th`ese, nous allons tout d'abord construire une nouvelle famille infinie de tri- angulations id´ealeset de H-triangulations pour la famille des nœuds twist hyperboliques. Ces triangulations donnent une nouvelle borne sup´erieurepour la complexit´ede Matveev des compl´ementaires des nœuds twist. Nous prouvons que les triangulations id´ealesci- dessus sont g´eom´etriquesen utilisant la technique de Futer{Gu´eritaud,`asavoir l'´etude de la fonctionnelle volume sur le poly`edredes structures d'angles. Nous utilisons en- suite ces triangulations pour calculer explicitement la fonction de partition de la TQFT de Teichm¨uller et prouver la conjecture du volume associ´eepour tous les nœuds twist hyperboliques `al'aide de la m´ethode du point col. Dans un deuxi`emetemps, nous pr´esentons des calculs et r´esultatssimilaires que pour les nœuds twist, mais cette fois pour une famille infinie de nœuds fibr´eshyperboliques dans les espaces lenticulaires L(n; 1) avec n > 1. Pour la triangulation id´eale,nous utilisons la triangulation monodromique de Floyd et Hatcher qui est g´eom´etrique.En ce qui concerne les H-triangulations, nous introduisons une m´ethode appel´ee T-chirurgie qui permet de construire des H-triangulations `apartir des triangulations id´ealesparticuli`eres. La th´eoriede Teichm¨ullerquantique a permis de construire des repr´esentations projectives unitaires du groupe de diff´eotopiesdes surfaces ´epoint´eesdans des espaces de Hilbert de dimension infinie. Il est esp´er´eque la trace des op´erateursquantiques associ´esdonne lieu `a des invariants de tores d'applications. La trace d'un op´erateurunitaire n'est pas toujours d´efinie.Cependant, nous verrons qu'il est possible de donner une interpr´etationde cette trace, dans le cas d'une monodromie pseudo-Anosov du tore ´epoint´e,`atravers la TQFT de Teichm¨ulleravec une certaine triangulation id´ealedu tore d'application munie d'une structure \presque" compl`ete. iii 概要 本論文には3つの新しい要素がある。最初にツイスト結び目のタイヒミュラー位相的量 子場の理論（TQFT）の分配関数の計算、次にレンズ空間の中に存在する、あるファイ バー結び目の無限例での計算、最後は量子タイヒミュラー理論においての指標の研究 である。 タイヒミュラーTQFTは2011年にアンデルセン（Andersen）とカシャエフ（Kashaev） によって構成された。これは四面体分割された結び目補空間の不変量であり、カシャ エフ不変量の無限次元版の結び目不変量でもある。 独自の体積予想を持ち、まず上 記2つの不変量の間に具体的な関係性があるとされ、そして結び目の体積がある漸近係 数として現れることが予想されている。 本論文では、まず、双曲ツイスト結び目に対して、新しい理想四面体分割とH-四面 体分割の無限例を構成する。これらの四面体分割は、ツイスト結び目補空間のマト ヴェーエフ複雑性の新しい上限を与える。 フュター·ゲリトー（Futer{Gu´eritaud）の 手法、すなわち、角度構造の多面体上の体積関数の研究を用いて、上記の理想四面体 分割が幾何学的であることを証明する。 そして、これらの四面体分割を用いて、タイ ヒミュラーTQFTの分配関数を明示的に計算し、すべての双曲ツイスト結び目につい て、鞍点法を用いて、関連する体積予想を証明する。 次いでツイスト結び目の場合と同様の計算と結果を提示するが、今回は、n > 1の 時のレンズ空間L(n; 1)での双曲ファイバー結び目の無限例に対しての計算である。 理想四面体分割には、すでに幾何学的だと知られている、フロイド（Floyd）とハッ チャー（Hatcher）によるモノドロミー四面体分割を使用する。H-四面体分割につい ては、T-手術という特別な理想四面体分割からH-四面体分割を作り出す手法を導入す る。 量子タイヒミュラー理論により、無限次元ヒルベルト空間上の穴あき曲面の写像類群 の射影表現を構成することが可能になった。この理論の量子作用素のトレースが写像 トーラス不変量であることが期待されている。本来、ユニタリ作用素のトレースは定 義されていない。しかし、穴あきトーラス上の擬アノソフモノドロミーの場合、タイ ヒミュラーTQFTを\ほぼ完全"な構造を持つ写像トーラスの、ある理想四面体分割で計 算することにより、このトレースを解釈することができるようになることを示す。 v Remerciements Cette th`esede doctorat ´etaitpour moi une grande aventure remplie d'enthousiasme et d'excitation, mais aussi parfois de moments difficiles. Notamment, l'ann´ee 2020 a, sans doute, ´et´eune ann´eetr`essp´eciale pour tout le monde. Quasiment tout est pass´e`adistance, et cela nous a permis de se rendre compte de l'importance de la solidarit´e. A cet ´egard, tous les obstacles rencontr´esdurant cette th`ese n'auraient pas pu ^etresurmont´essans le soutien de diverses personnes. Directement ou indirectement, elles ont toutes contribu´e`a une partie de cette th`ese.Par cons´equent, ces pages de remerciements leur sont d´edi´ees. Je voudrais tout d'abord exprimer ma plus grande reconnaissance `aRinat Kashaev, mon directeur de th`ese,sans qui l'aboutissement de cette th`esen'aurait pas ´et´epossible. Je le remercie pour sa disponibilit´e,sa patience, ses explications, ses conseils et pour m'avoir fait d´ecouvrirle monde de la recherche math´ematique. Exigent, mais il m'a toujours con- sid´er´ecomme un chercheur plut^otque comme un ´etudiant. Outre sa capacit´ecalculatoire exceptionnelle, j'admire aussi sa gentillesse et sa grande modestie, des qualit´eshumaines pr´ecieusesdans notre nouvelle section que je surnomme le \Kremlin helv´etique". Je me souviendrai comme il ´etaitimpressionn´ed'avoir vu une maid quand je lui ai fait visiter le quartier d'Akihabara apr`es la conf´erence`aWaseda en 2016. J'ai aussi aim´eles nom- breuses discussions non-math´ematiques (politique, Russie, sport, etc.) qu'on a eues dans son bureau. Enfin, merci aussi de m'avoir ordonn´ede m'inscrire `ala course de l'Escalade, qui est devenue depuis un ´ev´enement annuel essentiel pour moi. Je voudrais ensuite remercier Fathi Ben Aribi et Fran¸coisGu´eritaud,avec qui j'ai fait une grande partie de cette th`eseet qui ont ´et´eexemplaires pour la rigueur math´ematique.Mes remerciements vont ´egalement `aStavros Garoufalidis, Anders Karlsson et Alexis Virelizier pour avoir accept´ede faire partie des membres du jury. Je tiens aussi `aremercier nos secr´etairesqui ´etaient vraiment `al'´ecoutedes assistants : Annick, Nathalie et en particulier Joselle, qui s'est occup´eede tout mon dossier assez compliqu´e,qui m'a donn´ebeaucoup de stylos et qui m'a fait beaucoup rigoler.
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