Computer Bounds for Kronheimer-Mrowka Foam Evaluation

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 $J^{\mathrm{♯}}$, 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 $J^{\mathrm{♭}}$ for $J^{\mathrm{♯}}$. Of particular interest is the relationship between the dimension of $J^{\mathrm{♭}}(K)$ for a web $K$ and the number of Tait colorings $\mathrm{T}\mathrm{a}\mathrm{i}\mathrm{t}(K)$ 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 $J^{\mathrm{♭}}(K)$ 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{T}\mathrm{a}\mathrm{i}\mathrm{t}$(W_1) = 60$,butourresultssuggestthat$\dim J^\flat(W_1) = 58$.