  De Loera et al. 2009, showed that when the rank is fixed the Ehrhart
polynomial of a matroid polytope can be computed in polynomial time when the
number of elements varies. A key to proving this is the fact that the number of
simplicial cones in any triangulation of a tangent cone is bounded polynomially
in the number of elements when the rank is fixed. The authors speculated
whether or not the Ehrhart polynomial could be computed in polynomial time in
terms of the number of bases, where the number of elements and rank are allowed
to vary. We show here that for the uniform matroid of rank $r$ on $n$ elements,
the number of simplicial cones in any triangulation of a tangent cone is $n-2
\choose r-1$. Therefore, if the rank is allowed to vary, the number of
simplicial cones grows exponentially in $n$. Thus, it is unlikely that a
Brion-Lawrence type of approach, such as Barvinok's Algorithm, can compute the
Ehrhart polynomial efficiently when the rank varies with the number of
elements. To prove this result, we provide a triangulation in which the maximal
simplicies are in bijection with the spanning thrackles of the complete
bipartite graph $K_{r,n-r}$.
