## Abstract

Partially Poincaré-dualizing an embedded graph G on an arbitrary subset of edges was defined geometrically by Chmutov, using ribbon graphs. Part I of this series of papers introduced the partial-duality polynomial, which enumerates all the possible partial duals of the graph G, according to their Euler-genus, which can change according to the selection of the edge subset on which to dualize. Ellis-Monaghan and Moffatt have expanded the partial-duality concept to include the Petrie dual, the Wilson dual, and the two triality operators. Abrams and Ellis-Monaghan have given the five operators the collective name twualities. Part II of this series of papers derived formulas for partial-twuality polynomials corresponding to several fundamental sequences of embedded graphs. Here in Part III, we present an algorithm to calculate the partial-twuality polynomial of a ribbon graph G, for all twualities, which involves organizing the edge subsets of G into a hypercube and traversing that hypercube via a Gray code.

Original language | English |
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Pages (from-to) | 1119-1135 |

Number of pages | 17 |

Journal | Journal of Algebraic Combinatorics |

Volume | 54 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2021 |

### Bibliographical note

Funding Information:Thomas Tucker is supported by Simons Foundation Grant No. 317689.

Funding Information:

Jonathan Gross is supported by Simons Foundation Grant No. 315001.

Publisher Copyright:

© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Euler-genus
- Genus polynomial
- Gray code
- Partial duality
- Petrie dual
- Poincaré dual
- Ribbon graph
- Rotation system

## ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics