## Abstract

There are several different extensions of the Tutte polynomial to graphs embedded in surfaces. To help frame the different options, here we consider the problem of extending the Tutte polynomial to embedded graphs starting from first principles. We offer three different routes to defining such a polynomial and show that they all lead to the same polynomial. This resulting polynomial is known in the literature under a few different names including the ribbon graph polynomial, and 2-variable Bollobas-Riordan polynomial.

Our overall aim here is to use this discussion as a mechanism for providing a gentle introduction to the topic of Tutte polynomials for graphs embedded in surfaces.

Our overall aim here is to use this discussion as a mechanism for providing a gentle introduction to the topic of Tutte polynomials for graphs embedded in surfaces.

Original language | English |
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Title of host publication | 2023 MATRIX Annals |

Publisher | Springer |

Number of pages | 27 |

Publication status | Accepted/In press - 2024 |

### Publication series

Name | MATRIX Book Series |
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Publisher | Springer |

ISSN (Print) | 2523-3041 |

ISSN (Electronic) | 2523-305X |