The Plateau problem at infinity for horizontal ends and genus 1 Laurent Mazet 53A10minimal surfaceDirichlet problemboundary behaviourdegree theory 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. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2583 10.1512/iumj.2006.55.2583 en Indiana Univ. Math. J. 55 (2006) 15 - 64 state-of-the-art mathematics http://iumj.org/access/