Knot contact homology, string topology, and the cord algebra

Cieliebak, Kai; Ekholm, Tobias; Latschev, Janko; Ng, Lenhard

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

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 ${\Lambda }_K$, 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 ${\Lambda }_K$, and the Legendrian contact homology of ${\Lambda }_K$ 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 ${ℝ}^3\cup L_K$.

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

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