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The Tutte polynomials of genus $g$

Tsuyoshi Miezaki, Manabu Oura, Tadashi Sakuma, Hidehiro Shinohara

Abstract

In the paper [Proceedings of the Japan Academy, Ser. A Mathematical Sciences, 95(10) 111-113], the authors introduce the concept of the Tutte polynomials of genus $g$ and announce that each matroid $M$ can be reconstructed from its Tutte polynomial of genus $|\mathcal{B}(M)|$, where $\mathcal{B}(M)$ denotes the family of bases of $M$. In that paper, we also announced that, for all $g$, there exist inequivalent matroids that have the same Tutte polynomial of genus $g$. In this paper, we prove these theorems.

The Tutte polynomials of genus $g$

Abstract

In the paper [Proceedings of the Japan Academy, Ser. A Mathematical Sciences, 95(10) 111-113], the authors introduce the concept of the Tutte polynomials of genus and announce that each matroid can be reconstructed from its Tutte polynomial of genus , where denotes the family of bases of . In that paper, we also announced that, for all , there exist inequivalent matroids that have the same Tutte polynomial of genus . In this paper, we prove these theorems.
Paper Structure (13 sections, 33 theorems, 65 equations)

This paper contains 13 sections, 33 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary Properties on Matroids
  4. Matroid Base Graph
  5. Proof of Theorem \ref{['thm:main']} (1)
  6. Proof of Theorem \ref{['thm:main']} (2)
  7. Matroids $M,N$ such that $T^{(1)}(M)=T^{(1)}(N)$ and $T^{(2)}(M)\neq T^{(2)}(N)$
  8. Non-isomorphic matroids which need higher genus
  9. Concluding remarks
  10. Special values of $T^{(2)}$
  11. Witt's problem from the viewpoint of the Tutte polynomials of genus $g$
  12. Future problems
  13. The difference $T^{(2)}(R_{2n})-T^{(2)}(Q_{2n})$

Key Result

Theorem 1.7

Theorems & Definitions (71)

  • Example 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Oxley
  • ...and 61 more