  In this paper, we study Alexandrov-embedded r-noids with genus 1 and
horizontal ends. Such minimal surfaces are of two types and we build several
examples of the first one. We prove that if a polygon bounds an immersed
polygonal disk, it is the flux polygon of an r-noid with genus 1 of the first
type. We also study the case of polygons which are invariant under a rotation.
The construction of these surfaces is based on the resolution of the Dirichlet
problem for the minimal surface equation on an unbounded domain.
