For $n\geq 3$, an {$n$-spike is a matroid whose groundset can be partitioned into $n$ pairs called legs, such that the union of any two legs is both a circuit and a cocircuit. A spike may be extended by an element called a tip and/or coextended by an element called a cotip such that the union of the tip with each leg forms a triangle, and the union of the cotip with each leg forms a triad.

Spikes have arisen in many areas of matroid theory, sometimes notoriously, such as when Oxley, Vertigan & Whittle [1] used them to disprove Kahn's conjecture that the number of inequivalent representations of $3$-connected matroids over any finite field $GF(q)$ would be bounded above by some value $n(q)$.

I will aim this talk at a general combinatorial audience, spending some time on the widely-used geometrical representation of matroids and some of their properties. I will then present the characteristic polynomials, $\chi(M;\lambda)$, for the spike matroids (where the characteristic polynomial is a matroidal counterpart to the chromatic polynomial of a graph). I will also outline a proof that the zeros of these characteristic polynomials are bounded above by $\lambda=4$.

References

[1] Oxley, J., Vertigan, D., Whittle, G., On Inequivalent Representations of Matroids over Finite Fields. Journal of Combinatorial Theory, Series B 67, 325{343 (1996).