Скачать или смотреть Amit Kumar - HL Cone, Foams, and Graph Coloring

We begin with a review of the modern perspective on graph coloring which appeared in the work of Kronheimer-Mrowka and Khovanov-Robert. Next, we outline how the work of Treuman-Zaslow and Caslas-Zaslow lead to seeing graph coloring as topological defects labelled by the elements of Klein-Four Group. This highlights the quantum nature of graph coloring, namely, it satisfies the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with defects gives meaning to it. This TFT has the property that when evaluated on a planar trivalent graph, it provides the number of Tait-Coloring of it. Defects can be considered as a generalization of groups. With the Klein-four group as a 1-defect condition, we reinterpret graph coloring as sections of a certain cover, distinguishing a coloring (global-sections) from a coloring process (local-sections), and give a new formulation of some of Tait's work.