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The {\em rank $n$ swapping algebra} is a Poisson algebra defined on the setof ordered pairs of points of the circle using linking numbers, whose geometricmodel is given by a certain subspace of $(\mathbb{K}^n \times\mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$. For any idealtriangulation of $D_k$---a disk with $k$ points on its boundary, usingdeterminants, we find an injective Poisson algebra homomorphism from thefraction algebra generated by the Fock--Goncharov coordinates for$\mathcal{X}_{\operatorname{PGL}_n,D_k}$ to the rank $n$ swapping multifractionalgebra for $r=k\cdot(n-1)$ with respect to the (Atiyah--Bott--)Goldman Poissonbracket and the swapping bracket. This is the building block of the generalsurface case. Two such injective Poisson algebra homomorphisms related to twoideal triangulations $\mathcal{T}$ and $\mathcal{T}'$ are compatible with eachother under the flips.

Rank $n$ swapping algebra for $\operatorname{PGL}_n$ Fock-Goncharov $\mathcal{X}$ moduli space