On the structure of spikes

Spikes are an important class of 3-connected matroids. For an integer \(r\geq 3\), there is a unique binary r-spike denoted by \(Z_{r}\). 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.