Knot Contact Homology, String Topology, and the Cord Algebra

The conormal Lagrangian LK of a knotK 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 .