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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.
Wang et al. (Thu,) studied this question.