Knot contact homology, string topology, and the cord algebra
Date
2017
Authors
Journal Title
Journal ISSN
Volume Title
Repository Usage Stats
views
downloads
Citation Stats
Abstract
The conormal Lagrangian LKof a knot K in R3is the submanifold of the cotangent bundle T∗R3consisting 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 ΛKis a knot invariant of K, known as knot contact homology. We define a version of string topology for strings in R3∪ LKand 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∗R3with boundary on R3∪ LK.
Type
Department
Description
Provenance
Citation
Permalink
Published Version (Please cite this version)
Publication Info
Cieliebak, K, T Ekholm, J Latschev and L Ng (2017). Knot contact homology, string topology, and the cord algebra. Journal de l’École polytechnique — Mathématiques, 4. pp. 661–780. 10.5802/jep.55 Retrieved from https://hdl.handle.net/10161/17778.
This is constructed from limited available data and may be imprecise. To cite this article, please review & use the official citation provided by the journal.
Collections
Scholars@Duke

Lenhard Lee Ng
My research mainly focuses on symplectic topology and low-dimensional topology. I am interested in studying structures in symplectic and contact geometry (Weinstein manifolds, contact manifolds, Legendrian and transverse knots), especially through holomorphic-curve techniques. One particular interest is extracting topological information about knots through cotangent bundles, and exploring relations to topological string theory. I have also worked in Heegaard Floer theory, quantum topology, and sheaf theory, especially as they relate to Legendrian and transverse knots.
Unless otherwise indicated, scholarly articles published by Duke faculty members are made available here with a CC-BY-NC (Creative Commons Attribution Non-Commercial) license, as enabled by the Duke Open Access Policy. If you wish to use the materials in ways not already permitted under CC-BY-NC, please consult the copyright owner. Other materials are made available here through the author’s grant of a non-exclusive license to make their work openly accessible.