The ideal of relations for two generating sets of skein algebras S_n of the n-holed disk is generated by bounded-degree relations supported on subsurfaces with at most 6 holes (when q+q^{-1} invertible) or n holes (otherwise).
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SL(n) skein algebra of the twice punctured sphere is a commutative polynomial algebra in n-1 explicit crossing-free web generators for generic q.
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On skein algebras of planar surfaces
The ideal of relations for two generating sets of skein algebras S_n of the n-holed disk is generated by bounded-degree relations supported on subsurfaces with at most 6 holes (when q+q^{-1} invertible) or n holes (otherwise).
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Monomial web basis for the SL(N) skein algebra of the twice punctured sphere
SL(n) skein algebra of the twice punctured sphere is a commutative polynomial algebra in n-1 explicit crossing-free web generators for generic q.