Year: 2021
Type: article
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 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 Tait(W1)=60, but our results suggest that dimJ♭(W1)=58.
Source: Experimental Mathematics
Author: David Boozer
Institution: University of California, Los Angeles
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W2969861947