Noncrossing partitions of an annulus

The noncrossing partition poset associated to a Coxeter group $W$ and Coxeter element $c$ is the interval $[1,c{]}_T$ in the absolute order on $W$. We construct a new model of noncrossing partititions for $W$ of classical affine type, using planar diagrams (affine types $\tilde{A}$ and $\tilde{C}$ in this paper and affine types $\tilde{D}$ and $\tilde{B}$ in the sequel). The model in type $\tilde{A}$ consists of noncrossing partitions of an annulus. In type $\tilde{C}$, the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete $[1,c{]}_T$ to a lattice by factoring the translations in $[1,c{]}_T$, but the combinatorics of the planar diagrams leads us to make different choices about how to factor.