A new way to prove configuration reducibility using gauge theory

TL;DR

This paper introduces a gauge theory-based approach to prove configuration reducibility, providing a novel, non-computer method to verify the four color theorem through topological quantum field theory concepts.

Contribution

It presents a new gauge theory-inspired proof of the Birkhoff diamond's reducibility, offering an alternative to Kempe switch-based methods and proposing a potential non-computer proof of the four color theorem.

Findings

Proved the Birkhoff diamond is reducible using filtered homology.

Introduced the concept of state-reducible configurations.

Suggested gauge theory could lead to a non-computer proof of the four color theorem.

Abstract

We show how ideas coming out of gauge theory can be used to prove configurations in the list of ``633 unavoidable configurations" are reducible. In this paper, we prove the smallest nontrivial example, the Birkhoff diamond, is reducible using our filtered $3$- and $4$-color homology. This is a new proof of a 111-year-old result that is a direct consequence of a special (2+1)-dimensional topological quantum field theory. As part of the proof, we introduce the idea of a state-reducible configuration. Because state-reducibility does not involve Kempe switches, this leads to an independent way to verify the proof of the four color theorem. We conjecture that these gauge theoretic ideas could also lead to a non-computer-based proof of it.