Computer Bounds for Kronheimer-Mrowka Foam Evaluation

TL;DR

This paper explores a gauge-theoretic approach to the four color theorem via functors from webs and foams to vector spaces, using computer programs to analyze the dimensions related to Tait colorings.

Contribution

It introduces a computational method to analyze the functor $J^lat$ for webs, providing new insights into its dimension and its relation to Tait colorings for complex webs.

Findings

For the dodecahedral web, Tait colorings = 60, dimension of J^flat = 58.

Computer results strongly constrain possible dimensions of J^flat for various webs.

The approach offers a new perspective on the combinatorial properties of webs related to the four color theorem.

Abstract

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 $\dim J^\flat(W_1) = 58$.