Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension

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.