On skein algebras of planar surfaces

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 $n$-holed disk ${\mathrm{\Sigma }}_{0,n+1}$ over $R$. When $q+q^{-1}$ is invertible, in 2000 Przytycki and Sikora found a set of $n+\left(\genfrac{}{}{0px}{}{n}{2}\right)+\left(\genfrac{}{}{0px}{}{n}{3}\right)$ generators for ${\mathcal{S}}_n$; we show that the ideal of defining relations among these generators is generated by relations of degree $\le 6$ supported by certain subsurfaces diffeomorphic to ${\mathrm{\Sigma }}_{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, a set of $2^n-1$ generators for ${\mathcal{S}}_n$ was known to Bullock in 1999; we show that the ideal of defining relations is generated by relations of degree $\le 2k+2$ supported by certain subsurfaces diffeomorphic to ${\mathrm{\Sigma }}_{0,k+1}$ with $k\le n$. These results are substantial progresses towards answering Problem 1.92 (J) in the Kirby’s list.