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?
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.