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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| Veröffentlicht in: | Journal of graph theory Jg. 109; H. 4; S. 481 - 491 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. The total face color polynomial is based upon the Poincaré polynomials of a family of filtered ‐color homologies. It counts the number of ‐face colorings of ribbon graphs for each positive integer . As such, it may be seen as a successor of the Penrose polynomial, which at 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. |
| Author | McCarty, Ben Kauffman, Louis H. Baldridge, Scott |
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| Cites_doi | 10.37236/9214 10.1016/j.ejc.2022.103520 10.1142/S0218216525400024 10.1080/00029890.1975.11993805 10.1007/978-1-4614-6971-1 10.1006/jctb.1997.1752 10.1016/S0021-9800(66)80004-2 10.1007/s002080050030 10.1038/scientificamerican0476-126 10.1215/S0012-7094-00-10131-7 10.1016/j.ejc.2012.06.009 10.1007/978-94-009-0517-7_12 10.1016/j.aim.2004.10.015 |
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| References | 1966; 1 2005; 197 1997; 307 1997; 70 1990 2015; 60 2013; 34 2020; 27 1898; 5 1971 2025 2000; 101 2013 1975; 82 1976; 4 2022; 103 e_1_2_8_18_1 e_1_2_8_19_1 e_1_2_8_13_1 e_1_2_8_14_1 Kauffman L. H. (e_1_2_8_12_1) 2015; 60 e_1_2_8_16_1 e_1_2_8_3_1 e_1_2_8_2_1 e_1_2_8_5_1 Penrose R. (e_1_2_8_15_1) 1971 e_1_2_8_4_1 e_1_2_8_7_1 e_1_2_8_6_1 e_1_2_8_9_1 e_1_2_8_8_1 e_1_2_8_20_1 e_1_2_8_10_1 e_1_2_8_11_1 Petersen J. (e_1_2_8_17_1) 1898; 5 |
| References_xml | – volume: 307 start-page: 173 year: 1997 end-page: 189 article-title: The Penrose Polynomial of a Plane Graph publication-title: Mathematische Annalen – volume: 5 start-page: 225 year: 1898 end-page: 227 article-title: Sur le théorm̀e de Tait publication-title: L'Intermédiaire des Mathématiciens – volume: 1 start-page: 15 year: 1966 end-page: 50 article-title: On the Algebraic Theory of Graph Colorings publication-title: Journal of Combinatorial Theory – volume: 70 start-page: 166 issue: 1 year: 1997 end-page: 183 article-title: Tutte's Edge Coloring Conjecture publication-title: Journal of Combinatorial Theory, Series B – volume: 27 issue: 2 year: 2020 article-title: The 2‐Factor Polynomial Detects Even Perfect Matchings – volume: 34 start-page: 424 year: 2013 end-page: 445 article-title: A Penrose Polynomial for Embedded Graphs publication-title: European Journal of Combinatorics – volume: 4 start-page: 126 issue: 234 year: 1976 end-page: 130 article-title: Snarks, Boojums, and Other Conjectures Related to the Four‐Color‐Map Theorem publication-title: Mathematical Games, Scientific American – start-page: 123 year: 1990 end-page: 150 – volume: 197 start-page: 554 issue: 2 year: 2005 end-page: 586 article-title: An Endomorphism of the Khovanov Invariant publication-title: Advances in Mathematics – volume: 101 start-page: 359 issue: 3 year: 2000 end-page: 426 article-title: A Categorification of the Jones Polynomial publication-title: Duke Mathematical Journal – volume: 82 start-page: 630 year: 1975 end-page: 633 article-title: Infinite Families of Nontrivial Trivalent Graphs Which Are Not Tait Colorable publication-title: American Mathematical Monthly – year: 2025 article-title: Multi‐Virtual Knot Theory publication-title: Journal of Knot Theory and its Ramifications – year: 1971 – volume: 103 year: 2022 article-title: On Ribbon Graphs and Virtual Links publication-title: European Journal of Combinatorics – volume: 60 start-page: 251 issue: 1 year: 2015 end-page: 271 article-title: A State Calculus for Graph Coloring publication-title: Illinois Journal of Mathematics – year: 2013 – ident: e_1_2_8_2_1 – ident: e_1_2_8_5_1 doi: 10.37236/9214 – volume-title: Combinatorial Mathematics and Its Applications year: 1971 ident: e_1_2_8_15_1 – ident: e_1_2_8_4_1 doi: 10.1016/j.ejc.2022.103520 – ident: e_1_2_8_13_1 doi: 10.1142/S0218216525400024 – ident: e_1_2_8_20_1 doi: 10.1080/00029890.1975.11993805 – ident: e_1_2_8_11_1 – ident: e_1_2_8_14_1 doi: 10.1007/978-1-4614-6971-1 – ident: e_1_2_8_3_1 – volume: 60 start-page: 251 issue: 1 year: 2015 ident: e_1_2_8_12_1 article-title: A State Calculus for Graph Coloring publication-title: Illinois Journal of Mathematics – ident: e_1_2_8_19_1 doi: 10.1006/jctb.1997.1752 – volume: 5 start-page: 225 year: 1898 ident: e_1_2_8_17_1 article-title: Sur le théorm̀e de Tait publication-title: L'Intermédiaire des Mathématiciens – ident: e_1_2_8_18_1 doi: 10.1016/S0021-9800(66)80004-2 – ident: e_1_2_8_8_1 doi: 10.1007/s002080050030 – ident: e_1_2_8_16_1 doi: 10.1038/scientificamerican0476-126 – ident: e_1_2_8_6_1 doi: 10.1215/S0012-7094-00-10131-7 – ident: e_1_2_8_9_1 doi: 10.1016/j.ejc.2012.06.009 – ident: e_1_2_8_10_1 doi: 10.1007/978-94-009-0517-7_12 – ident: e_1_2_8_7_1 doi: 10.1016/j.aim.2004.10.015 |
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| Snippet | 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... The total face color polynomial is based upon the Poincaré polynomials of a family of filtered ‐color homologies. It counts the number of ‐face colorings of... 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... |
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| SubjectTerms | coloring Graph coloring Graphs Homology perfect matching Polynomials Quantum theory spectral sequence Sums Tensors topological quantum field theory |
| Title | A State Sum for the Total Face Color Polynomial |
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