Is the instanton homology for webs and foams a categorified Chern-Simons?

Question

In their paper, Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my limited understanding, $J^{\#}$ is a sort of a TQFT (with $\mathbb{F}_2$ coefficient), or a functor from the category of webs and foams to the category of $\mathbb{F}_2$ vector spaces and morphisms.

There is a familiar TQFT in one lower dimension which is clearly related to the number of Tait colorings : namely, the (Chern-Simons) quantum invariant for the second symmetric representation of $SU(2)$. In the classical limit $q=1$, it is exactly the number of Tait colorings.

I know this is probably just a far-fetched speculation, but I am curious if these two TQFTs can be somehow related. More precisely, is it possible to categorify the latter such that it becomes $J^{\#}$ when reduced to the $\mathbb{F}_2$ coefficient?

Answer 1

It’s not clear what you mean by categorify here: the J# invariant doesn’t have a grading in their definition. Just taking dimension to get a number then doesn’t seem to yield a polynomial invariant.

However, they have a variation of this invariant for SU(3) representations whose dimension yields the number of Tait colorings of cubic graphs. It is bigraded, so has an Euler characteristic (Kronheimer discusssed this at the Georgia Topology conference last summer). The natural conjecture is that this might be related to the Penrose number, which is a signed count of Tait colorings. The exact triangle for J# (which should also exist for SU(3)) involves three graphs which also relate the Penrose number. However, the Penrose number is only defined up to sign (depending on a framing), whereas I think the SU(3) invariant should have an absolute grading. So such a relation might be too much to expect.