  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.
