  This paper is the third in a series of manuscripts that examine the
combinatorics of the Kunz polyhedron $P_m$, whose positive integer points are
in bijection with numerical semigroups (cofinite subsemigroups of $\mathbb
Z_{\ge 0}$) whose smallest positive element is $m$. The faces of $P_m$ are
indexed by a family of finite posets (called Kunz posets) obtained from the
divisibility posets of the numerical semigroups lying on a given face. In this
paper, we characterize to what extent the minimal presentation of a numerical
semigroup can be recovered from its Kunz poset. In doing so, we prove that all
numerical semigroups lying on the interior of a given face of $P_m$ have
identical minimal presentation cardinality, and we provide a combinatorial
method of obtaining the dimension of a face from its corresponding Kunz poset.
