Math @ Duke

Publications [#330521] of Lenhard L. Ng
Papers Published
 Cieliebak, K; Ekholm, T; Latschev, J; Ng, L, Knot contact homology, string topology, and the cord algebra, vol. 4
(January, 2017),
pp. 661780 [doi]
(last updated on 2018/01/18)
Abstract: 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 Λ K , and the Legendrian contact homology of Λ K is a knot invariant of K, known as knot contact homology. We define a version of string topology for strings in R 3 ∪ 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 ∪ L K .


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