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