Some Hard Problems on Matroid Spikes

Spikes form an interesting class of 3-connected matroids of branch-width 3. We show that some computational problems are hard on spikes with given matrix representations over infinite fields. Namely, the question whether a given spike is the free spike is co-NP-hard (though the property itself is definable in monadic second-order logic); and the task to compute the Tutte polynomial of a spike is #P-hard (even though that can be solved efficiently on all matroids of bounded branch-width which are represented over a finite field).