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Knot contact homology, string topology, and the cord algebra

Publication ,  Journal Article
Cieliebak, K; Ekholm, T; Latschev, J; Ng, L
Published in: Journal de l'Ecole Polytechnique - Mathematiques
January 1, 2017

The conormal Lagrangian LK of a knot K in R3 is the submanifold of the cotangent bundle T∗R3 consisting of covectors along K that annihilate tangent vectors to K. By intersecting with the unit cotangent bundle S∗R3, one obtains the unit conormal ΛK, and the Legendrian contact homology of ΛK is a knot invariant of K, known as knot contact homology. We define a version of string topology for strings in R3 ∪ LK 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∗R3 with boundary on R3 ∪ LK.

Duke Scholars

Published In

Journal de l'Ecole Polytechnique - Mathematiques

DOI

EISSN

2270-518X

ISSN

2429-7100

Publication Date

January 1, 2017

Volume

4

Start / End Page

661 / 780
 

Citation

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Cieliebak, K., Ekholm, T., Latschev, J., & Ng, L. (2017). Knot contact homology, string topology, and the cord algebra. Journal de l’Ecole Polytechnique - Mathematiques, 4, 661–780. https://doi.org/10.5802/jep.55
Cieliebak, K., T. Ekholm, J. Latschev, and L. Ng. “Knot contact homology, string topology, and the cord algebra.” Journal de l’Ecole Polytechnique - Mathematiques 4 (January 1, 2017): 661–780. https://doi.org/10.5802/jep.55.
Cieliebak K, Ekholm T, Latschev J, Ng L. Knot contact homology, string topology, and the cord algebra. Journal de l’Ecole Polytechnique - Mathematiques. 2017 Jan 1;4:661–780.
Cieliebak, K., et al. “Knot contact homology, string topology, and the cord algebra.” Journal de l’Ecole Polytechnique - Mathematiques, vol. 4, Jan. 2017, pp. 661–780. Scopus, doi:10.5802/jep.55.
Cieliebak K, Ekholm T, Latschev J, Ng L. Knot contact homology, string topology, and the cord algebra. Journal de l’Ecole Polytechnique - Mathematiques. 2017 Jan 1;4:661–780.

Published In

Journal de l'Ecole Polytechnique - Mathematiques

DOI

EISSN

2270-518X

ISSN

2429-7100

Publication Date

January 1, 2017

Volume

4

Start / End Page

661 / 780