Catenoid limits of singly periodic minimal surfaces with scherk-type ends

We construct families of embedded, singly periodic minimal surfaces of any genus g in the quotient with any even number 2n > 2 of almost parallel Scherk ends. A surface in such a family looks like n parallel planes connected by n - 1 + g small catenoid necks. In the limit, the family converges to an n-sheeted vertical plane with n - 1 + g singular points, termed nodes, in the quotient. For the nodes to open up into catenoid necks, their locations must satisfy a set of balance equations whose solutions are given by the roots of Stieltjes polynomials.