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Call an ordered $N$-tuple of points in the Euclidean plane ${\mathbb R} ^2$ "in general position" if no three points of the points in the set are collinear. As a function of $N$ how many components are there for the space of such generic $N$-tuples?

DATA: For $N =3$ I get 2 components, corresponding to the sign of the area of the triangle whose vertices are the 3 points. For $N=4$ I get 14 (though I may have miscounted).

MOTIVATION: My motivation came from a mechanics question which led to trying to understand the topology of these individual components and then being told by M. Kapovich that Mnev's Universality Theorem asserts their topologies can be essentially arbitrary. (!!)

ATTEMPTED BOUND: Write out all possible signs of areas of all the triangles $\Delta(i,j,k)$ formed by triples of vertices $q_i, q_j, q_k$ from our generic configuration $(q_1, q_2, \ldots, q_N)$. There are $2^D$ with $D = {{N \choose 3}}$ such signs. So that's my "bound". Unfortunately, this "bound" seems to be neither lower or upper. It is not a lower bound since $2^4 = 16 > 14$. Moreover, it is almost certainly not an upper bound, since by Mnev again, as $N$ gets large one finds that for some of these sign choices ("generic combinatorial types") the corresponding set of generic points itself falls into (arbitrarily!) many components.

REFORMULATION? This question, like Mnev's theorem, can be reformulated in the language of "realizations of (oriented) matroids". The choice of signs of areas above seems to be a "chirotope". I would be happy for some expert to do a clean job of a matroid reformulation of my question and happier still if they could point me towards an answer, even for $N = 5,6, 7, 8,9, 10$.

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  • $\begingroup$ I think what you're after is related to the number of "order types" ist.tugraz.at/aichholzer/research/rp/triangulations/ordertypes $\endgroup$
    – j.c.
    Commented Jul 18, 2017 at 0:06
  • $\begingroup$ Nevermind, since your point sets are ordered, the enumeration will be different. There's also potentially the issue mentioned in this question, where mathoverflow.net/questions/98283 , though the answer doesn't seem to give a definitive answer for point configurations in general position. $\endgroup$
    – j.c.
    Commented Jul 18, 2017 at 0:27
  • $\begingroup$ Thank you j.c. for the references!. The tables here at least seem to yield lower bounds. Perusing the `Handbooks'' I found a table on p. 147 of a chapter titled PSEUDOLINE ARRANGEMENTS'' by Stefan Felsner and Jacob E. Goodman which apparently asserts a lower bound for the number of combinatorial types of the form $2^{4N log N}$ but these are not simple''. I am not sure though if they are talking about the same thing that I am. $\endgroup$
    – Richard Montgomery
    Commented Jul 18, 2017 at 3:22
  • $\begingroup$ Maybe the question is easier if the N ordered generic points lie in the real projective plane instead of the affine plane? Then for N=4 we get only 1 component... $\endgroup$
    – Richard Montgomery
    Commented Jul 18, 2017 at 5:01
  • $\begingroup$ MR1350697 M. Kapovich, and J. Millson, On the moduli space of polygons in the Euclidean plane. J. Differential Geom. 42 (1995), no. 1, 133–164. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 18, 2017 at 7:57

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