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.