Genus one minimal k-noids and saddle towers in ${\mathbb{H}}^2\times \mathbb{R}$

For each $k\ge 3$, we construct a 1-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space ${\mathbb{H}}^2\times \mathbb{R}$ 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 ${\mathbb{H}}^2\times \mathbb{R}$ 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 also provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of ${\mathbb{H}}^2\times \mathbb{R}$ by the action of a hyperbolic or parabolic translation.