In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key counterexample in hyperplane arrangement theory. In particular, this implies that the topology of the complement manifold of a hyperplane arrangement is not determined by the combinatorics of its lattice of intersections, i.e, its underlying matroid.

I am interested in some SageMath computations and I like to know an explicit presentation of the matroid introduced by Rybnikov. Maybe this matroid is already available in the SageMath library.

$\textbf{Question:}$ What are the ground set, the rank and the bases set $\mathfrak{B}$ of the Rybnikov matroid?

  • $\begingroup$ Rybnikov calls this matroid MacLane matroid, and specifies how it is constructed from the affine plane of order 3. $\endgroup$ May 28 '19 at 16:06
  • $\begingroup$ @DimaPasechnik: I believe the OP wants data on the matroid associated with the point configuration $C_{13}$ and not the MacLane matroid associated with $C_8$. $\endgroup$
    – Aaron Dall
    Jun 10 '19 at 5:44
  • 1
    $\begingroup$ Indeed, this might be the case, sorry. Anyhow, people are most welcome to contrubute to Sagemath library of matroids... :) $\endgroup$ Jun 10 '19 at 12:15

Since this is still open, I'll describe how to find an answer.

You have a presentation of the matroid as an arrangement of hyperplanes. Now trace through the 'cryptomorphisms' described in the Wikipedia article (see also Oxley's book, etc):

  • Any hyperplane arrangement has a intersection lattice, which consists of all intersections of hyperplanes, and which is usually taken to be ordered by reverse inclusion. So the whole space is the $\hat{0}$ element, and the hyperplanes are the atoms.

  • A nice enough hyperplane arrangement has a geometric lattice as its intersection lattice.

  • A geometric lattice of height r is associated with a matroid. The elements are the atoms of the lattice (so, the hyperplanes in the arrangement in this situation). In general, the elements of the lattice are the flats in the matroid, and the matroid has rank r.

  • The bases of the matroid are thus those sets of atoms of cardinality r whose join is the top element $\hat{1}$ of the intersection lattice.

You mention computation with SageMath, and I expect that you can probably use the same to automate the calculation necessary in these steps.


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