On the structure of spikes

Spikes are an important class of 3-connected matroids. For an integer $r\geq 3$,thereisauniquebinaryr-spikedenotedby$Z_{r}$. When a circuit-hyperplane of $Z_r$ is relaxed, we obtain another spike and repeating this procedure will produce other non-binary spikes. The $es$-splitting operation on a binary spike of rank $r$, may not yield a spike. In this paper, we give a necessary and sufficient condition for the $es$-splitting operation to construct $Z_{r+1}$ directly from $Z_r$. Indeed, all binary spikes and many of non-binary spikes of each rank can be derived from the spike $Z_3$ by a sequence of The $es$-splitting operations and circuit-hyperplane relaxations.