Small remarkable matroids

Question

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) associated to a matroid $M$ with a ground set $\sharp(E)\leq7,$ without minors of Fano or dual-Fano type.

I also wrote a sage programme which explitly compute this group. I've already do the computations for the matroids:

$F_{7}^{-},$ $O7,$ $P6,$ $P7,$ $Q6$ $R6.$

In fact, these are all the matroids with $\sharp(E)\leq7$ I've found in the sage library. My questions are the following:

1) Are there some other interesting matroids $M$ with the condition $\sharp(E)\leq7$ (without minors of Fano or dual-Fano type)?

2) Maybe this matroids arise from some interesting geometric or topological construction. Could you please explain where these matroids come from?

Answer 1

Other than those you've listed, the most obvious "interesting" matroids I can think of on up to 7 elements are:

• $M(K_4)$, the cycle matroid of $K_4$,
• $W^3$, the "whirl" obtained from $M(K_4)$ by relaxing a 3-element circuit,
• $U_{r,n}$, the rank-r uniform matroid of size n.
• $(F_7^-)^*$, the dual of the non-Fano

I'm pretty sure that your list plus $M(K_4)$, $W^3$, $U_{2,6}$, $U_{4,6}$, and $U_{3,6}$ covers all the 3-connected 6-element matroids. For 7 elements, there are many other matroids but I don't think there are any more that are interesting enough to have standard names.