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3 thoughts on “Q&A

  1. Question from Luis Goddyn

    Question for y’all,
    Here is a partial list of complete matroid classes (closed under minors, duals, direct sums, principal extensions). Each class in the list is properly contained in its predecessors. (In fact, M(K_4) is a minimal excluded minor for each class.)
    1. M(K_4)-free
    2. Base orderable
    3. k-base orderable (k=2,3,…)
    4. Strongly base orderable
    5. Gammoids
    Now, every gammoid is orientable (indeed R-representable).
    Q1. Does another class in this list consist entirely of orientable (or R-representable) matroids?
    Q2. Is there a non-orientable matroid without M(K_4) minor?
    I have not found much helpful literature so far.

    • Hi Luis, interesting questions!

      I don’t know enough about orientable matroids to have proper intuition. I can give an answer to Q1 part (ii) (the answer is no) and for what it’s worth, I think the answer to Q2 will be yes.

      Take AG(2,3), the ternary affine plane. The appendix in Oxley’s book tells us that it has no M(K_4)-minor. It is not real-representable, so no class in your list contains only real-representable matroids.

      On the other hand, AG(2,3) is base-orderable, and therefore orientable. So I used Sage to construct ternary coextensions of AG(2,3) that do not have M(K_4), in the hope that some of them will be non-orientable. Here’s an example, represented over GF(3):

      [1 2 1 2 1 2 1 2 1 0]
      [0 0 1 1 2 2 1 1 0 0]
      [1 1 1 1 1 1 0 0 0 0]
      [0 0 1 1 0 1 2 0 2 1]

      But now I’m stuck, because I don’t know how to quickly test whether this is non-orientable.



      EDIT: I got the order of subclass inclusion wrong, so of course AG(2,3)
      tells us nothing about any of the classes other than Class (1), the
      matroids with no M(K_4)-minor. This class certainly contains
      matroids that are not real-representable.

      I can now do better than that, and show that Class (1) contains
      non-orientable matroids. In their paper Orientability of
      , Bland and Las Vergnas call the following matroid the
      MacLane matroid, and refer to the paper Representation of
      , by Ingleton. Ingleton in turn refers to the 1936 paper
      Some interpretations of abstract linear dependence in terms of
      projective geometry
      by Saunders MacLane, but MacLane credits the
      configuration to Friedrich Levi, in Geometrische
      , of 1931!

      In any case, I entered the matroid into Sage with the following command:


      Bland and Las Vergnas state that M is an excluded minor for the class of
      orientable matroids (p. 114). On the other hand, if I ask Sage:


      it reports


      So M is an example of a M(K_4)-free matroid that is non-orientable.

      • UPDATE: Luis pointed out that the MacLane matroid is actually just $AG(3,2)\backslash e$. So this matroid is not orientable. On the other hand, Lemma (4.4) in Oxley’s paper “A characterisation of the ternary matroids with no $M(K_{4})$-minor” says that a ternary rank-$3$ matroid is strongly base-orderable if and only if it has no $M(K_{4})$-minor. Since this property holds for $AG(3,2)\backslash e$, it is strongly base-orderable, and not orientable. So the answer to Luis’s Q1 is no: every one of his classes, other than the class of gammoids, contains non-orientable matroids.

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