  Kronheimer and Mrowka recently suggested a possible approach towards a new
proof of the four color theorem that does not rely on computer calculations.
Their approach is based on a functor $J^\sharp$, which they define using gauge
theory, from the category of webs and foams to the category of vector spaces
over the field of two elements. They also consider a possible combinatorial
replacement $J^\flat$ for $J^\sharp$. Of particular interest is the
relationship between the dimension of $J^\flat(K)$ for a web $K$ and the number
of Tait colorings $\mathrm{Tait}(K)$ of $K$; these two numbers are known to be
identical for a special class of "reducible" webs, but whether this is the case
for nonreducible webs is not known. We describe a computer program that
strongly constrains the possibilities for the dimension and graded dimension of
$J^\flat(K)$ for a given web $K$, in some cases determining these quantities
uniquely. We present results for a number of nonreducible example webs. For the
dodecahedral web $W_1$ the number of Tait colorings is $\mathrm{Tait}(W_1) =
60$, but our results suggest that $\dim J^\flat(W_1) = 58$.
