On skein algebras of planar surfaces

Abstract: Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$. Let $\mathcal{S}n$ denote the Kauffman bracket skein algebra of the planar surface $\Sigma{0,n+1}$ over $R$. When $q+q^{-1}$ is invertible in $R$, we find a generating set for $\mathcal{S}n$, and show that the ideal of defining relations is generated by relations of degree at most $6$ supported by certain subsurfaces homeomorphic to $\Sigma{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, we find another generating set for $\mathcal{S}_n$, and show that the ideal of defining relations is generated by certain relations of degree at most $2n+2$.