## 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 (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.).

Original language | English (US) |
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Pages (from-to) | 615-630 |

Number of pages | 16 |

Journal | Experimental Mathematics |

Volume | 32 |

Issue number | 4 |

DOIs | |

State | Published - 2023 |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

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