Homeomega14⁺

omega14⁺ with line F projected to infinity

Richter-Gebert, [1996] presents two oriented matroids, both derived from a third. His title, two interesting oriented matroids, suggests he sees the third one as dull — we include it nevertheless. The basic derivation is from this matrix:

A B C D E F G H I J K L M N
1 0 1 0 1 1 2 3 2 3 1 1 -1 0
0 1 0 1 1 2 1 2 3 1 3 -1 1 0
0 0 1 1 2 2 2 4 4 4 4 4 4 1

This oriented matroid is defined with a chirotope being the 3 by 3 determinants of the above matrix except that:

χ(L,M,N) = +1

This oriented matroid is interesting, because like Suvorov's one, its realization space is disconnected. It is also easier to define, using only integers in the matrix.

Reoriented

B, D

Line Order At Infinity

A, D< G< H, C< J< I, E, B< L< N< K, M

Bases

Positively Oriented

ADG, ADH, ADC, ADJ, ADI, ADE, ADB, ADL, ADN, ADK, ADM, AGI, AGK, AHI, AHK, ACJ, ACI, ACE, ACK, ACM, AJI, AJE, AJK, AEK, ABL, ABN, ABK, ABM, ALN, ALK, ALM, ANK, ANM, DHI, DCI, DJI, DBL, DBK, DLK, DNK, GHI, GHK, GCJ, GCI, GCE, GCL, GCN, GCK, GCM, GJI, GJE, GJN, GJK, GJM, GEK, GBL, GBN, GBK, GBM, GLN, GLK, GLM, GNK, GNM, HCJ, HCI, HCE, HCB, HCL, HCN, HCK, HCM, HJI, HJE, HJL, HJN, HJK, HJM, HEK, HBL, HBN, HBK, HBM, HLN, HLK, HLM, HNK, HNM, CJI, CBL, CBN, CBK, CBM, CLN, CLK, CLM, CNK, CNM, JBL, JBN, JBK, JBM, JLN, JLK, JLM, JNK, JNM, IEL, IEN, IEK, IEM, IBL, IBN, IBK, IBM, ILN, ILK, ILM, INK, INM, EBL, EBN, EBK, EBM, ELN, ELK, ELM, ENK, ENM, BNK, LNK, LNM,

Negatively Oriented

AGC, AGJ, AGB, AGL, AGN, AGM, AHC, AHJ, AHB, AHL, AHN, AHM, ACB, ACL, AJB, AJL, AJN, AIE, AIB, AIL, AIN, AIM, AEB, AEL, AEN, AEM, AKM, DGH, DGC, DGJ, DGE, DGB, DGL, DGN, DGK, DGM, DHC, DHJ, DHE, DHB, DHL, DHN, DHK, DHM, DCB, DCL, DCN, DCM, DJB, DJL, DJN, DJM, DIE, DIB, DIL, DIN, DIK, DIM, DEB, DEL, DEN, DEM, DBM, DLN, DLM, DNM, DKM, GHC, GHJ, GHB, GHL, GHN, GHM, GJB, GIE, GIB, GIL, GIN, GIK, GIM, GEB, GEL, GEN, GEM, GKM, HIE, HIB, HIL, HIN, HIK, HIM, HEB, HEL, HEN, HEM, HKM, CJB, CJL, CJN, CJM, CIE, CIB, CIL, CIN, CIK, CIM, CEB, CEL, CEN, CEM, CKM, JIE, JIB, JIL, JIN, JIK, JIM, JEB, JEL, JEN, JEM, JKM, IKM, EKM, BLN, BLM, BNM, BKM, LKM, NKM,

Not Bases

AGH, AGE, AHE, ACN, AJM, AIK, DGI, DCJ, DCE, DCK, DJE, DJK, DEK, DBN, GHE, GCB, GJL, HJB, CJE, CJK, CEK, JEK, IEB, BLK,

The Pseudolines