  The $k$-associahedron $Ass_k(n)$ is the simplicial complex of
$(k+1)$-crossing-free subgraphs of the complete graph with vertices on a
circle. Its facets are called $k$-triangulations.
  We explore the connection of $Ass_k(n)$ with the Pfaffian variety
$Pf_k(n)\subset {\mathbb K}^{\binom{[n]}2}$ of antisymmetric matrices of rank
$\le 2k$.
  First, we characterize the Gr\"obner cone $Grob_k(n)\subset{\mathbb
R}^{\binom{[n]}2}$ producing as initial ideal of $I(Pf_k(n))$ the
Stanley-Reisner ideal of $Ass_k(n)$ (that is, the monomial ideal generated by
$(k+1)$-crossings). This implies that $k$-triangulations are bases in the
algebraic matroid of $Pf_k(n)$, a matroid closely related to low-rank
completion of antisymmetric matrices.
  We then look at the tropicalization of $Pf_k(n)$ and show that $Ass_k(n)$
embeds naturally as the intersection of $\operatorname{trop}(Pf_k(n))$ and
$Grob_k(n)$, and is contained in the totally positive part
$\operatorname{trop}^+( Pf_k(n))$ of it.
  We show that for $k=1$ and for each triangulation $T$ of the $n$-gon, the
projection of this embedding of $Ass_k(n)$ to the $n-3$ coordinates
corresponding to diagonals in $T$ gives a complete polytopal fan, realizing the
associahedron. This fan is linearly isomorphic to the $\mathbf g$-vector fan of
the cluster algebra of type $A$, shown to be polytopal by Hohlweg, Pilaud and
Stella in (2018).
