A State Sum for the Total Face Color Polynomial

ABSTRACT The total face color polynomial is based upon the Poincaré polynomials of a family of filtered n‐color homologies. It counts the number of n‐face colorings of ribbon graphs for each positive integer n. As such, it may be seen as a successor of the Penrose polynomial, which at n = 3 counts 3...

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Bibliographic Details
Published in:Journal of graph theory Vol. 109; no. 4; pp. 481 - 491
Main Authors: Baldridge, Scott, Kauffman, Louis H., McCarty, Ben
Format: Journal Article
Language:English
Published: Hoboken Wiley Subscription Services, Inc 01.08.2025
Subjects:
coloring
Graph coloring
Graphs
Homology
perfect matching
Polynomials
Quantum theory
spectral sequence
Sums
Tensors
topological quantum field theory
ISSN:0364-9024, 1097-0118
Online Access:Get full text
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Summary:ABSTRACT The total face color polynomial is based upon the Poincaré polynomials of a family of filtered n‐color homologies. It counts the number of n‐face colorings of ribbon graphs for each positive integer n. As such, it may be seen as a successor of the Penrose polynomial, which at n = 3 counts 3‐edge colorings (and consequently 4‐face colorings) of planar trivalent graphs. In this paper, we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.
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ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.23239