Tsukamoto, [2013] shows that Suvorov's oriented matroid is not the smallest possible one with disconnected realization space. His examples use 13 elements instead of Suvorov's and Richter-Gerbert's 14.
The basic approach is the same as Richter-Gerbert, take a 3 by n matrix, define a chirotope function using the 3 by 3 determinants, carefully select one triple for which that chirotope is zero, where the oriented matroid can be locally perturbed, and define two new oriented matroids by perturbing in either direction.
Here is Tsukamoto's matrix, which is evaluated with s=½, t=½ and u=⅓.
| A | B | C | D | E | F | G | H | I | J | K | L | M |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0 | 0 | 1 | s | s | 0 | 1 | 1 | st | s+t-u-st+su | s+t-st-s2u | s(t-u+su) |
| 0 | 1 | 0 | 1 | 0 | 1 | t | t | u | t | t-u + su | t | t-u+su |
| 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1-su | 1-u+su | 1-su | 1-u+su |
This oriented matroid has chirotope defined by the above matrix, except that:
Tsukamoto shows that the realization space is disconnected. That means that there are two different realizations of this pseudoline arrangement as equivalent line arrangements, such that it is not possible to continuously transform one to the other through a sequence of equivalent line arrangements.
A, D, H, I
K, E, D< C< I< M< G< J, A< H< F< B
