Computer Bounds for Kronheimer–Mrowka Foam Evaluation

Abstract

Kronheimer and Mrowka recently suggested a possible approach toward a new proof of the four color theorem. Their approach is based on a functor $J^{♯}$, 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 $J^{♭}$ for $J^{♯}$. Of particular interest is the relationship between the dimension of $J^{♭}(K)$ for a web K and the number of Tait colorings $\text{Tait}(K)$ 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 $J^{♭}(K)$ 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 W₁ the number of Tait colorings is $\text{Tait}(W_1)=60$, but our results suggest that $\dim J^{♭}(W_1)=58$.

KEYWORDS:

• Webs
• foams
• trivalent graphs
• Tait colorings
• four-color theorem

Acknowledgments

The author would like to express his gratitude toward Ciprian Manolescu for providing invaluable guidance, and towards Andrea Bertozzi for the use of the UCLA Joshua computing cluster. The author would also like to thank Mikhail Khovanov, Peter Kronheimer, Tomasz Mrowka, and Louis-Hadrien Robert for providing many helpful comments on an earlier version of this paper that led to significant changes in the current version.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author was partially supported by NSF (grant number DMS-1708320).