The type and cardinality of minimal presentations of numerical semigroups with embedding dimension four

For a numerical semigroup $S$ with embedding dimension four, we study the relationship between its type $t(S)$ and the cardinality of its minimal presentations $\eta (S)$. Using an approach based on the geometry of the Apéry set, we prove that $4t(S)+5\ge \eta (S)\ge t(S)-11$. This resolves a problem of Moscariello and Sammartano asking whether $t(S)$ is bounded by a function of $\eta (S)$, and improves the previously known bound $9t(S)+4\ge \eta (S)$ due to Bresinsky.