A complete knot invariant from contact homology

Abstract

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant. arXiv:1606.07050v3 [math.SG] 25 Sep 2017Of these, R KK is precisely the degree 0 knot contact homology of K and contains a subring Z[l ±1 , m ±1 ] once we equip K with an orientation and framing (which we choose to be the Seifert framing), where l, m denote the longitude and meridian of K; R Kp and R pK are left and right modules, respectively, over R KK . We remark that (LCH * ) Λ K ,Λ K , (LCH * ) Λ K ,Λp , Proposition 4.4. There are isomorphismswhere the first line is a ring isomorphism, and the second and third lines send the left and right actions of Cord KK to the left and right actions of R KK . Under these isomorphisms, the map µ :Proof. The first line is proved in Proposition 2.9 of [CELN17], and the other two lines have the same proof. The fact that these isomorphisms preserve multiplication follows formally from the construction of the cord algebra and modules. The description of µ as a concatenation product is a direct consequence of Proposition 3.11.

Knot Contact Homology and Open Gromov–witten Theory

“…Our first result states that the contact deformation class of Λ K encodes the isotopy class of K. Let p ∈ R 3 be a point not on K and let Λ p ⊂ ST * R 3 denote the Legendrian conormal sphere of p. We consider certain filtered quotients of CE(Λ K ∪ Λ p ), called R Kp , R pK , and R KK , together with a product operation m : R Kp ⊗ R pK → R KK , borrowed from wrapped Floer cohomology. A version of this theorem was first proved by Shende [28] using micro-local sheaves and was reproved using holomorphic disks in [13]. We point out that the Legendrian conormal tori of any two knots are smoothly isotopic when considered as ordinary submanifolds of ST * R 3 .…”

“…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 .…”

“…The knot contact homology of the previous section can also be interpreted, via Legendrian surgery, in terms of partially wrapped Floer cohomology that in turn is connected to the micro-local sheaves used by Shende [28] to prove the completness result in Theorem 1.1. We give a very brief discussion and refer to [14,Section 6] for more details.…”

Generating families and augmentations for Legendrian surfaces

“…The synergy between microlocal sheaf invariants and J-holomorphic curves for conormal tori is also nicely illustrated in the exciting recent result, due to Vivek Shende, that knots in R 3 are determined up to isotopy by the Legendrian isotopy type of their conormal tori [29]. Shende's first proof of this result made use of microlocal sheaf theory and was then followed with a second proof by Ekholm, Ng, and Shende using methods of J-holomorphic curve theory [13].…”

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.…”