KW Semigroups -- Their Betti Numbers, Ap\'ery Posets and Tangent Cones

Let p<q be coprime integers. Kunz-Waldi semigroups are numerical semigroups containing p and q and contained in <p,q,r>, where 2r=p,q,p+q whichever is even. In this paper, we prove a conjecture on the Betti numbers of the semigroup rings of these semigroups, showing that they coincide with those of the ideal of 2x2 minors of a 2xn generic matrix, where n is the embedding dimension. Moreover, we characterize the Ap\'ery posets of Kunz-Waldi semigroups and determine when their tangent cones are Cohen-Macaulay.