  We study the $k$-th nearest neighbor distance function from a finite
point-set in $\mathbb{R}^d$. We provide a Morse theoretic framework to analyze
the sub-level set topology. In particular, we present a simple
combinatorial-geometric characterization for critical points and their indices,
along with detailed information about the possible changes in homology at the
critical levels. We conclude by computing the expected number of critical
points for a homogeneous Poisson process. Our results deliver significant
insights and tools for the analysis of persistent homology in order-$k$
Delaunay mosaics, and random $k$-fold coverage.
