  $k$-point crossover operators and their recombination sets are studied from
different perspectives. We show that transit functions of $k$-point crossover
generate, for all $k>1$, the same convexity as the interval function of the
underlying graph. This settles in the negative an open problem by Mulder about
whether the geodesic convexity of a connected graph $G$ is uniquely determined
by its interval function $I$. The conjecture of Gitchoff and Wagner that for
each transit set $R_k(x,y)$ distinct from a hypercube there is a unique pair of
parents from which it is generated is settled affirmatively. Along the way we
characterize transit functions whose underlying graphs are Hamming graphs, and
those with underlying partial cube graphs. For general values of $k$ it is
shown that the transit sets of $k$-point crossover operators are the subsets
with maximal Vapnik-Chervonenkis dimension. Moreover, the transit sets of
$k$-point crossover on binary strings form topes of uniform oriented matroid of
VC-dimension $k+1$. The Topological Representation Theorem for oriented
matroids therefore implies that $k$-point crossover operators can be
represented by pseudosphere arrangements. This provides the tools necessary to
study the special case $k=2$ in detail.
