The Betti Numbers of Kunz-Waldi Semigroups

Given two coprime numbers $p<q$, KW semigroups contain $p,q$ and are contained in $⟨p,q,r⟩$ where $2r=p,q,p+q$ whichever is even. These semigroups were first introduced by Kunz and Waldi. Kunz and Waldi proved that all $KW$ semigroups of embedding dimension $n\ge 4$ have Cohen-Macaulay type $n-1$ and first Betti number $\left(\genfrac{}{}{0px}{}{n}{2}\right)$. In this paper, we characterize KW semigroups whose defining ideal is generated by the $2\times 2$ minors of a $2\times n$ matrix. In addition, we identify all KW semigroups that lie on the interior of the same face of the Kunz cone ${\mathcal{C}}_p$ as a KW semigroup with determinantal defining ideal. Thus, we provide an explicit formula for the Betti numbers of all those KW semigroups.