On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere

Based on the presentation of Kauffman bracket skein algebra thickened torus given by third author in previous work [4], Frohman and Gelca established a complete description multiplicative operation leading to famous product-to-sum formula. In this paper, we study structure four-holed sphere. We present an algorithm compute product any two elements algebra, give explicit formula for some families curves. surmise that has quasi-polynomial growth with respect number crossings pair Further, conjecture existence positive basis algebra.