Monomial web basis for the SL(N) skein algebra of the twice punctured sphere

For any non-zero complex number q, excluding finitely many roots of unity of small order, a linear basis for the SL (n) skein algebra of the twice punctured sphere is constructed. In particular, the skein algebra is a commutative polynomial algebra in n-1 generators, where each generator is represented by an explicit SL (n) web, without crossings, on the surface. This includes the case q=1, where the skein algebra is identified with the coordinate ring of the SL (n) character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called SL (n) quantum trace map, due originally to Bonahon-Wong in the case n=2. A consequence of the proof is that the polynomial algebra sits as a distinguished subalgebra of the L\^e-Sikora SL (n) stated skein algebra of the annulus. We end by discussing the relationship with Fock-Goncharov duality.