Knot contact homology, string topology, and the cord algebra


Journal Article

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.

Full Text

Duke Authors

Cited Authors

  • Cieliebak, K; Ekholm, T; Latschev, J; Ng, L

Published Date

  • January 1, 2017

Published In

Volume / Issue

  • 4 /

Start / End Page

  • 661 - 780

Electronic International Standard Serial Number (EISSN)

  • 2270-518X

International Standard Serial Number (ISSN)

  • 2429-7100

Digital Object Identifier (DOI)

  • 10.5802/jep.55

Citation Source

  • Scopus