HL Cone, Foams, and Graph Coloring

DOI

10.2991/978-94-6463-789-2_2How to use a DOI?

Keywords

Tait coloring; topological field theory; defects

Abstract

We begin with a review of the modern perspective on graph coloring, which appeared in the work of many people, including Penrose. Next, we outline how the work of Treuman-Zaslow and Caslas-Zaslow led to graph coloring being seen as topological defects labeled by the elements of the Klein-Four Group. The TFT with defects has the property that when evaluated on a planar trivalent graph, it provides the number of Tait-Colorings of it. The same construction can be generalized to arbitrary groups and a presentation, translating the word problem to a cobordism problem. The connection between graph coloring and the word problem results in a hypothesis that translates Tait-colorability to the solvability of the word problem in the Klein-Four Group.

Copyright

© 2025 The Author(s)

Open Access

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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TY  - CONF
AU  - Amit Kumar
PY  - 2025
DA  - 2025/07/17
TI  - HL Cone, Foams, and Graph Coloring
BT  - Proceedings of the International Conference on Knots, Quivers and Beyond (ICKQB 2025)
PB  - Atlantis Press
SP  - 4
EP  - 18
SN  - 2352-541X
UR  - https://doi.org/10.2991/978-94-6463-789-2_2
DO  - 10.2991/978-94-6463-789-2_2
ID  - Kumar2025
ER  -