Knot contact homology, string topology, and the cord algebra

2016/01/09 by Cieliebak, Kai, Ekholm, Tobias, Latschev, Janko +1
#53D42 #55P50 #57M27 #57R17 #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1601.02167

Abstract

The conormal Lagrangian L_K 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 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 ℝ³ ∪ 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^* ℝ³ with boundary on ℝ³ ∪ L_K.

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