Computer Bounds for Kronheimer–Mrowka Foam Evaluation

Kronheimer and Mrowka recently suggested a possible approach toward a new proof of the four color theorem. Their approach is based on a functor (Formula presented.), which they define using gauge theory, from the category of webs and foams to the category of F-vector spaces, where F is the field of two elements. They also consider a possible combinatorial replacement (Formula presented.) for (Formula presented.). Of particular interest is the relationship between the dimension of (Formula presented.) for a web K and the number of Tait colorings (Formula presented.) 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 (Formula presented.) 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 (Formula presented.), but our results suggest that (Formula presented.).