Knot contact homology, string topology, and the cord algebra

Kai Cieliebak,Janko Latschev,Lenhard Ng,Tobias Ekholm

doi:10.5802/jep.55

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 .

Highlights

To a smooth n-manifold Q we can naturally associate a symplectic manifold and a contact manifold: its cotangent bundle T ∗Q with the canonical symplectic structure ω = dp ∧ dq, and its unit cotangent bundle S∗Q ⊂ T ∗Q with its canonical contact structure ξ = ker(p dq)
We present a different approach to knot contact homology and the cord algebra via string topology
Let N be a tubular neighborhood of K. For this definition we do not need a framing for the knot K; later, when we identify H0string(K) with the cord algebra, it will be convenient to fix a framing, which will in turn fix an identification of N with S1 × D2

Summary

Introduction

— Besides being more natural from the viewpoint of string homology, the stipulation that λ, μ do not commute with cords (in the cord algebra) or Reeb chords (in the DGA) is essential for our construction, in Section 2.4 below, of a map from degree 0 homology to the group ring of π, the fundamental group of the knot complement This in turn is what allows us to (re)prove that knot contact homology detects the unknot, among other things. There is a cord formulation for modified string homology H0string(K) (as introduced at the end of Section 2.1), along the lines of Definition 2.6: this is A /I , where A is the non-unital algebra generated by nonempty products of cords (the difference from A being that A does not contain words of the form λaμb, which have no cords), and I is the ideal of A generated by skein relations (ii) through (iv) from Definition 2.6, without (i). – Morse-theoretical arguments on the space Σlin to prove that Φ induces an isomorphism on degree zero homology

Holomorphic functions near corners

String homology in arbitrary degree

The chain map from Legendrian contact homology to string homology

Proof of the isomorphism in degree zero

Transversely cut out solutions and orientations