  The Kauffman bracket skein algebra is a quantization of the algebra of
regular functions on the $SL_2$ character variety of a topological surface. We
realize the skein algebra of the $4$-punctured sphere as the output of a mirror
symmetry construction based on higher genus Gromov-Witten theory and applied to
a complex cubic surface. Using this result, we prove the positivity of the
structure constants of the bracelets basis for the skein algebras of the
$4$-punctured sphere and of the $1$-punctured torus. This connection between
topology of the $4$-punctured sphere and enumerative geometry of curves in
cubic surfaces is a mathematical manifestation of the existence of dual
descriptions in string/M-theory for the $\mathcal{N}=2$ $N_f=4$ $SU(2)$ gauge
theory.
