On the structure of spikes II

The class spike is an important class of 3-connected matroids. For an integer r≥3, each matroid that is obtained by relaxing one of the circuit-hyperplanes of an r-spike (spike with rank r) is isomorphic to another r-spike and repeating this procedure will produce other r-spikes. So it will be useful to characterize the collection of circuit-hyperplanes of a spike. In this paper, we first introduce a technique to characterize the only two ternary r-spikes in terms of their matrix representations, circuit-hyperplanes, and (r+1)-circuits and then we generalize this technique to characterize the class of r-spikes that is representable over the field GF(p), for all primes p. Moreover, we pose a conjecture which would characterize the class of all r-spikes by using all GF(p)-representable r-spikes.