  In this paper we give a topological interpretation and diagrammatic calculus
for the rank $(n-2)$ Askey-Wilson algebra by proving there is an explicit
isomorphism with the Kauffman bracket skein algebra of the $(n+1)$-punctured
sphere. To do this we consider the Askey-Wilson algebra in the braided tensor
product of $n$ copies of either the quantum group
$\mathcal{U}_q{(\mathfrak{sl}_2)}$ or the reflection equation algebra. We then
use the isomorpism of the Kauffman bracket skein algebra of the
$(n+1)$-punctured sphere with the $\mathcal{U}_q{(\mathfrak{sl}_2})$ invariants
of the Aleeksev moduli algebra to complete the correspondence. We also find the
graded vector space dimension of the $\mathcal{U}_q{(\mathfrak{sl}_2})$
invariants of the Aleeksev moduli algebra and apply this to finding a
presentation of the skein algebra of the five-punctured sphere and hence also
find a presentation for the rank $2$ Askey-Wilson algebra.
