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PreprintCC BY 4.0arXiv

A face polynomial invariant for embedded graphs

  • Hongxia Ma
  • Metrose Metsidik
Open at sourcePDFDOICite

Abstract

We introduce the face polynomial of a ribbon graph, a generating function for face numbers under the ribbon group action. It addresses the face-count case of the Ellis--Monaghan--Moffatt question on twisted duality. We derive deletion--contraction recursions and prove that this polynomial coincides with the vertex polynomial for ⟨δ⟩, ⟨δτ⟩, and ⟨δ,τ⟩. We characterize monomial cases and obtain formulas and recursions for Euler-genus partial-twuality polynomials. For orientable graphs, we show that the partial-dual face polynomial has both parities whenever an edge exists, so the partial-dual Euler-genus polynomial has only even exponents.

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Cite

BibTeXRIS
BibTeX
@misc{ma2026face,
  title = {{A face polynomial invariant for embedded graphs}},
  author = {Hongxia Ma and Metrose Metsidik},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05090},
  eprint = {2611.05090},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05090},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A face polynomial invariant for embedded graphs
AU  - Hongxia Ma
AU  - Metrose Metsidik
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05090
UR  - https://arxiv.org/abs/2611.05090
AB  - We introduce the face polynomial of a ribbon graph, a generating function for face numbers under the ribbon group action. It addresses the face-count case of the Ellis--Monaghan--Moffatt question on twisted duality. We derive deletion--contraction recursions and prove that this polynomial coincides with the vertex polynomial for ⟨δ⟩, ⟨δτ⟩, and ⟨δ,τ⟩. We characterize monomial cases and obtain formulas and recursions for Euler-genus partial-twuality polynomials. For orientable graphs, we show that the partial-dual face polynomial has both parities whenever an edge exists, so the partial-dual Euler-genus polynomial has only even exponents.
ER  -