What is knot contact homology?

Question

Recently, it was conjectured by the paper of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed $A$-polynomial of a knot $K$ can be obtained by finding the difference equation of minimal order for the colored HOMFLY polynomials of the knot $K$. This conjecture seems to hold true for torus knots and twist knots. However, I do not understand what the knot contact homology is.

First of all, the knot contact homology describes knot invariants as invariants of the Legendrian submanifolds in the contact manifold. A knot is realized by an intersection of the cosphere bundle $ST^∗M$ of a 3-manifold $M$ with the unit conormal bundle $\Lambda_K$ where $ST^∗M$ admits a contact structure.

1) Is there any way to visualize an intersection of $ST^∗M$ with $\Lambda_K$?

The knot contact homology is constructed by the Legendrian differential graded algebra (DGA)

2) Why do you need DGA to obtain homology theory invariant under Legendrian isotopy?

From the combinatorial definition (Appendix B of the paper), I cannot see the reason why this is isomorphic to Legendrian DGA. Although the differentials are determined by the braiding data of a knot, it seems to me that there is no contact structure involved.

3) Could the isomorphism between the two DGA be explained in layman's terms?

I do not understand what the augmentation polynomials of the knot contact homology are.

4) Is there any relation between augmentation polynomials and Porincare-Chekanov polynomials?

In addition,

5) I would like to know if there is an explicit connection of knot contact homology to other knot homologies such as Khovanov-Rozansky and HOMFLY homology.

Answer 1

Check out these lecture notes (and the references listed therein) http://www.renyi.hu/~cast2012/ng-cast.pdf

1) Visualize elements of $ST^{*}\mathbb{R}^{3}$ as unit length vectors -- not necessarily based at the origin -- in $\mathbb{R}^{3}$. The projection $\pi$ onto $\mathbb{R}^{3}$ send each vector to its basepoint. The unit conormal bundle bundle $\Lambda_{K}$ consists of those vectors orthogonal to $K$. If you apply the time-$\epsilon$ Reeb flow (=geodesic flow) to $\Lambda_{K}$ for $0<\epsilon$ very small and then apply $\pi$, you will get the boundary of a tubular neighborhood of $K$ in $\mathbb{R}^{3}$.

2) I take it that this question means ``Why can't you define homology groups by counting Reeb chords and holomorphic disks (which are easier to work with than DGAs)?'' You can only define homology groups by counting holomorphic strips (disks with 2 boundary punctures) if you can say that the only way that strips break (in a Gromov limit) is into a pair of strips. This if often possible for Lagrangian intersection problems by imposing various geometric constraints which rule out undesireable degenerations of homolorphic disks. In this Legendrian case, this does not work if you are to obtain a Legendrian isotopy class invariant. Play around with some examples of Legendrian knots as in Etnyre-Ng-Sullivan's paper and this will become clear.

3) Not really: (Mild) You have to first identify $ST^{*} \mathbb{R}^{3}$ with the 1-jet space $ J^{1}S^{2}=\mathbb{R}\times T^{*}S^{2}$ of $S^{2}$. This isn't too hard to intuit (the zero section of the 1-jet space is identified with $\pi^{-1}$ of the origin in $\mathbb{R}^{3}$). (Medium) Now think about what the conormal bundle $\Lambda_{\mu}$ of the unknot $\mu$ looks like in this 1-jet space. As an approximation, the projection of this torus onto $S^{2}$ is obtained by applying the geodesic flow to a fiber of the unit tangent bundle of $S^{2}$ for time $t\in[0,2\pi]$. (Spicy) If $K$ is braided about $\mu$, then $\Lambda_{K}$ will be braided about $\Lambda_{\mu}$ in $J^{1}S^{2}$. It's well-known that any knot is braided about the unknot. (Habanero) Use a direct limit argument and Ekholm's Morse flow trees technique to read off the holomorphic disks from the braiding data.

4) They're both defined by looking at the representations of the DGAs.

5) Ng's papers describe how knot contact homology encodes various knot polynomials. It is conjectured that you can use it to obtain the HOMFLY-PT polynomial. As for knot homology theories, I'm sure that people have thoughts on this but as far as I'm aware no conjectures have been anounced.