Genus 1 minimal k-noids and saddle towers in H-2 x R Castro Infantes, Jesús Minimal surfaces Finite total curvature Minimal k-noids Saddle towers Conjugate construction The authors would like to express their gratitude to Magdalena Rodriguez for her valuable comments during the preparation of this manuscript, as well as to the anonymous referee for the thorough revision of the manuscript, which has greatly improved the final presentation. This research was supported by MINECO-FEDER project MTM2017-89677-P and by MCIN/AEI project PID2019-111531GA-I00. The first author is also supported by the FPU programme from MICINN and by MCIN/AEI project PID2020-117868GB-I00. The second author is also supported by EBM/FEDER UJA 2020 project 1380860. For each k >= 3, we construct a 1-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space H-2 x R with genus 1 and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus 1 and 2k ends in the quotient of H-2 x R by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature -4k pi. Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus 1 in quotients of H-2 x R by the action of a hyperbolic or parabolic translation. 2022-01-31T11:29:39Z 2022-01-31T11:29:39Z 2022-01-06 journal article Castro-Infantes, J., & Manzano, J. (2022). GENUS 1 MINIMAL k-NOIDS AND SADDLE TOWERS IN H-2 x R. Journal of the Institute of Mathematics of Jussieu, 1-21. doi:[10.1017/S1474748021000591] http://hdl.handle.net/10481/72557 10.1017/S1474748021000591 eng http://creativecommons.org/licenses/by/3.0/es/ open access Atribución 3.0 España Cambridge University Press