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The Kauffman bracket skein algebra is a quantization of the algebra ofregular functions on the $SL_2$ character variety of a topological surface. Werealize the skein algebra of the $4$-punctured sphere as the output of a mirrorsymmetry construction based on higher genus Gromov-Witten theory and applied toa complex cubic surface. Using this result, we prove the positivity of thestructure constants of the bracelets basis for the skein algebras of the$4$-punctured sphere and of the $1$-punctured torus. This connection betweentopology of the $4$-punctured sphere and enumerative geometry of curves incubic surfaces is a mathematical manifestation of the existence of dualdescriptions in string/M-theory for the $\mathcal{N}=2$ $N_f=4$ $SU(2)$ gaugetheory.

Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus