  The Kauffman bracket skein algebra of a surface is a generalization of the
Jones polynomial invariant for links and plays a principal role in the
Witten-Reshetikhin- Turaev topological quantum field theory. However, the
multiplicative structure of the skein algebra is not well understood, with a
priori exponential complexity. We consider the case of one-hole torus, and
provide a polynomial algorithm for computing multiplication of any two skein
elements. Some closed form formulas for multiplication of curves with low
crossing number are also given.
