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Matroids with different configurations and the same $\mathcal{G}$-invariant

Joseph E. Bonin

Abstract

From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a newer, stronger invariant, the $\mathcal{G}$-invariant. To gauge how much additional information the configuration contains compared to these invariants, it is of interest to have methods for constructing matroids with different configurations but the same $\mathcal{G}$-invariant. We offer several such constructions along with tools for developing more.

Matroids with different configurations and the same $\mathcal{G}$-invariant

Abstract

From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a newer, stronger invariant, the -invariant. To gauge how much additional information the configuration contains compared to these invariants, it is of interest to have methods for constructing matroids with different configurations but the same -invariant. We offer several such constructions along with tools for developing more.

Paper Structure

This paper contains 5 sections, 14 theorems, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation, background, and prior results
  3. The tools
  4. Application: a construction modifying a lattice
  5. Application: a construction using paving matroids

Key Result

Lemma 2.1

Let $M$ be a matroid with neither loops nor coloops. If $X$ is any cyclic flat of $M$, then $\mathcal{Z}(M|X)$ is the interval $[\emptyset,X]$ in $\mathcal{Z}(M)$, and so the lattice $\mathcal{Z}(M/X)$ is isomorphic to the interval $[X,E(M)]$ in $\mathcal{Z}(M)$.

Figures (6)

  • Figure 1: Two rank-$3$ matroids, $M$ and $N$, and their lattices of cyclic flats.
  • Figure 2: The configurations of the matroids $M$ and $N$ in Figure \ref{['fig:runningmatroid']}. The pairs shown are $(s(x),\rho(x))$, the size and rank of the corresponding cyclic flat.
  • Figure 3: A lattice $L$ and two extension of the type in the construction treated in Section \ref{['sec:lattice']}, where $L_s$ is isomorphic to $\mathcal{Z}(M)$ and where $L_t$ is isomorphic to $\mathcal{Z}(N)$ for the matroids $M$ and $N$ in Figure \ref{['fig:runningmatroid']}.
  • Figure 4: The lattice in Example \ref{['ex:1']}.
  • Figure 5: The rank-$3$ paving matroids in Example \ref{['ex:3']}.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:simplify']}
  • Example 1
  • Theorem 3.2
  • Lemma 3.3
  • Corollary 3.4
  • ...and 15 more