  We construct minimal surfaces by gluing simply periodic Karcher-Scherk saddle
towers along their wings. Such constructions were previously implemented
assuming a horizontal reflection plane. We break this symmetry by prescribing
phase differences between the saddle towers. It turns out that, in addition to
the previously known horizontal balancing condition, the saddle towers must
also be balanced under a subtle vertical interaction. This interaction vanishes
in the presence of a horizontal reflection plane, hence was not perceived in
previous works.
  Our construction will be presented in a series of papers. In this first paper
of the series, we will explain the background of the project and establish the
graph theoretical setup that will be useful for all papers in the series. The
main task of the current paper is to glue saddle towers into triply periodic
minimal surfaces (TPMSs). Our construction expands many previously known TPMSs
into new 5-parameter families, therefore significantly advances our knowledge
on the space of TPMSs.
