Knot contact homology, string topology, and the cord algebra

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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.





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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

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Lenhard Lee Ng

Professor of Mathematics

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.

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