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Kai Cieliebak; Tobias Ekholm; Janko Latschev; Lenhard Ng
Knot contact homology, string topology, and the cord algebra
(Homologie de contact pour les nœuds, topologie des cordes et algèbre des cordes)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 661-780, doi: 10.5802/jep.55
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Class. Math.: 53D42, 55P50, 57R17, 57M27
Keywords: Holomorphic curve, string topology, conormal bundle, knot invariant, Lagrangian submanifold, Legendrian submanifold

Résumé - Abstract

The conormal Lagrangian $L_K$ of a knot $K$ in $\mathbb{R}^{3}$ is the submanifold of the cotangent bundle $T^{\ast }\mathbb{R}^{3}$ consisting of covectors along $K$ that annihilate tangent vectors to $K$. By intersecting with the unit cotangent bundle $S^{\ast }\mathbb{R}^{3}$, 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 $\mathbb{R}^{3}\cup L_K$ 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 $T^{\ast }\mathbb{R}^{3}$ with boundary on $\mathbb{R}^{3}\cup L_K$.

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