Knot contact homology, string topology, and the cord algebra

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 . 1 arXiv:1601.02167v2 [math.SG] 23 May 2017 * (K).In its most general form (see [11,33]), knot contact homology is the homology of a differential graded algebra over the group ringwhere the images of λ, µ under the connecting homomorphism generate H 1 (Λ K ) = H 1 (T 2 ) and U generates H 2 (S * R 3 ). The isomorphism class of H contact * (K) as a Z[λ ±1 , µ ±1 , U ±1 ]-algebra is then an isotopy invariant of the framed oriented knot K.The topological content of knot contact homology has been much studied in recent years; see for instance [1] for a conjectured relation, which we will not discuss here, to colored HOMFLY-PT polynomials and topological strings. One part of knot contact homology that has an established topological interpretation is its U = 1 specialization. In [31,32], the third author constructed a knot invariant called the cord algebra Cord(K), whose definition we will review in Section 2.2. The combined results of [31,32,15] then prove that the cord algebra is isomorphic as a Z[λ ±1 , µ ±1 ]-algebra to the U = 1 specialization of degree 0 knot contact homology. We will assume throughout this paper that we have set U = 1; 1 then the result is: 1 ([31, 32, 15]). H contact 0 (K) ∼ = Cord(K).It has been noticed by many people that the definition of the cord algebra bears a striking resemblance to certain operations in string topology [4,36]. Indeed, Basu, McGibbon, Sullivan, and Sullivan used this observation in [2] to construct a theory called "transverse string topology" associated to any codimension 2 knot K ⊂ Q, and proved that it determines the U = λ = 1 specialization of the cord algebra.In this paper, we present a different approach to knot contact homology and the cord algebra via string topology. Motivated by the general picture sketched by the 1 However, we note that it is an interesting open problem to find a similar topological interpretation of the full degree 0 knot contact homology as a Z[λ ±1 , µ ±1 , U ±1 ]-algebra. 2 In the presence of contractible closed geodesics in Q, this will require augmentations by holomorphic planes in T * Q, see e.g. [6].

Knot Contact Homology and Open Gromov–witten Theory

“…In this section we define a topological model for knot contact homology in low degrees that one can think of as the string topology of a certain singular space. Our treatment will be brief and we refer to [3,13] for full details. Let K ⊂ R 3 be a knot and p ∈ R 3 − K a point with Lagrangian conormals L K and L p .…”

“…is the chain parameterized by the locus in σ of strings with components in S 3 that intersect K at interior points. The operation splits the curve at such intersection and inserts a spike in L K , see [3]. The operation δ Q K is defined similarly exchanging the role of R 3 and L K .…”

Floer Homologies, with Applications

“…Chekanov and Eliashberg used them to distinguish Legendrian knots with the same "classical" invariants [38]. They also give rise to very fine invariants of smooth knots in Ê 3 , see [40,61,148]. Embedded contact homology is due to Hutchings.…”

A complete knot invariant from contact homology

“…Other evidence in this direction is provided by the fact that knot contact homology recovers enough of the knot group (the fundamental group of the knot complement) to detect the unknot [Ng08] and torus knots, among others [GL]. These results use the ring structure on the "fully noncommutative" version of knot contact homology, where the algebra is generated by Reeb chords along with homology classes in H 1 (Λ K ) that do not commute with Reeb chords; see [Ng14,CELN17]. However, the question of whether knot contact homology is a complete invariant remains open.…”

“…In this paper, we extend the isomorphism from [CELN17] to show that the KCH-triple can also be computed using broken strings. Using this presentation, we prove a ring isomorphism…”