I’m going to try and fill in the fudge factors. There is a unique way of doing this, given known results and assuming the only allowed constants are the Thurston-Bennequin number and the number of components. First let me assemble the known results.
We take out of [HR]:
Definition. For a ruling
The ruling polynomial is by definition
Example. The standard unknot has a single ruling, and the corresponding disk has Euler characteristic
Aside. This sort of genus expansion is entirely standard in string theory; see e.g. any equation in the paper where Ooguri and Vafa made the Ooguri-Vafa conjecture. They all have exactly this exponent, although the curves being counted there are holomorphic whereas the curves here are Lagrangian. Given that in this case as well we are computing the HOMFLY polynomial (even with very similar correction factors, see the introduction to [STZ]) it seems like there should be in some cases some other version of the topological string which explains this coincidence, or perhaps some way to “just make a hyperkahler rotation” although we have never succeeded in making actual sense of such a rotation.
The main result of [HR] is:
Theorem. [HR]. Let
Moreover, if we’re counting ungraded augmentations, introduce the broken Euler characteristic
where
Example. The DGA of the unknot has a single generator, which has degree 1. So
Remark. In [HR], they leave off the
On the other hand, in [STZ], we showed that for rainbow braid closures,
Here,
Thus the desired conjecture, normalized, is:
Conjecture.
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