Knot contact homology, string topology, and the cord algebra

Kai Cieliebak; Tobias Ekholm; Janko Latschev; Lenhard Ng

doi:10.5802/jep.55

Knot contact homology, string topology, and the cord algebra
[Homologie de contact pour les nœuds, topologie des cordes et algèbre des cordes]

Kai Cieliebak ¹ ; Tobias Ekholm ^2,³ ; Janko Latschev ⁴ ; Lenhard Ng ⁵

¹ Institut für Mathematik, Universität Augsburg 86135 Augsburg, Germany
² Department of Mathematics, Uppsala University 751 06 Uppsala, Sweden and
³ Institut Mittag-Leffler Aurav 17, 182 60 Djursholm, Sweden
⁴ Universität Hamburg, Fachbereich Mathematik Bundesstraße 55, 20146 Hamburg, Germany
⁵ Department of Mathematics, Duke University Durham, NC 27708-0320, USA

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 661-780.

Résumé
Abstract

Le fibré conormal lagrangien $L_{K}$ d’un nœud $K$ dans $ℝ^{3}$ est la sous-variété du fibré cotangent $T^{*} ℝ^{3}$ formée des covecteurs le long de $K$ qui annulent les vecteurs tangents à $K$ . En l’intersectant avec le fibré cotangent unitaire $S^{*} ℝ^{3}$ , on obtient le fibré conormal unitaire $Λ_{K}$ , dont l’homologie de contact legendrienne est un invariant du nœud $K$ , appelé homologie de contact pour les nœuds. Nous définissons une version de la topologie des cordes pour des cordes dans $ℝ^{3} \cup L_{K}$ et montrons qu’elle est isomorphe en degré $0$ à l’homologie de contact pour les nœuds. La topologie des cordes permet une approche topologique de l’algèbre des cordes (qui est aussi isomorphe à l’homologie de contact pour les nœuds en degré $0$ ) et la relie au groupe du nœud. Ceci donne, joint à cet isomorphisme, une nouvelle démonstration du fait que l’homologie de contact pour les nœuds détecte le nœud trivial. Nos techniques font intervenir une analyse détaillée de certains espaces de modules de disques holomorphes dans $T^{*} ℝ^{3}$ avec bord dans $ℝ^{3} \cup L_{K}$ .

The conormal Lagrangian $L_{K}$ of a knot $K$ in $ℝ^{3}$ is the submanifold of the cotangent bundle $T^{*} ℝ^{3}$ consisting of covectors along $K$ that annihilate tangent vectors to $K$ . By intersecting with the unit cotangent bundle $S^{*} ℝ^{3}$ , one obtains the unit conormal $Λ_{K}$ , and the Legendrian contact homology of $Λ_{K}$ is a knot invariant of $K$ , known as knot contact homology. We define a version of string topology for strings in $ℝ^{3} \cup L_{K}$ and prove that this is isomorphic in degree $0$ to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree $0$ knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in $T^{*} ℝ^{3}$ with boundary on $ℝ^{3} \cup L_{K}$ .

Reçu le : 2016-02-04
Accepté le : 2017-05-22
Publié le : 2017-06-08

MR Zbl

DOI : 10.5802/jep.55

Classification : 53D42, 55P50, 57R17, 57M27
Keywords: Holomorphic curve, string topology, conormal bundle, knot invariant, Lagrangian submanifold, Legendrian submanifold
Mots-clés : Courbe holomorphe, topologie des cordes, fibré conormal, invariant de nœud, sous-variété lagrangienne, sous-variété legendrienne

Affiliations des auteurs :

Kai Cieliebak ¹ ; Tobias Ekholm ^{2, 3} ; Janko Latschev ⁴ ; Lenhard Ng ⁵

¹ Institut für Mathematik, Universität Augsburg 86135 Augsburg, Germany
² Department of Mathematics, Uppsala University 751 06 Uppsala, Sweden and
³ Institut Mittag-Leffler Aurav 17, 182 60 Djursholm, Sweden
⁴ Universität Hamburg, Fachbereich Mathematik Bundesstraße 55, 20146 Hamburg, Germany
⁵ Department of Mathematics, Duke University Durham, NC 27708-0320, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Knot contact homology, string topology, and~the cord algebra},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     publisher = {\'Ecole polytechnique},
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Kai Cieliebak; Tobias Ekholm; Janko Latschev; Lenhard Ng. Knot contact homology, string topology, and the cord algebra. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 661-780. doi : 10.5802/jep.55. https://jep.centre-mersenne.org/articles/10.5802/jep.55/

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