Fast Algorithm for Multiplication on the Skein Algebra of One-hole Torus

Sike Wang; Helen Wong

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arXiv · 2405.20159

Fast Algorithm for Multiplication on the Skein Algebra of One-hole Torus

Sike Wang·Helen Wong

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Abstract

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.

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Sike Wang, Helen Wong. 2024-05-30. Fast Algorithm for Multiplication on the Skein Algebra of One-hole Torus. https://arxiv.org/abs/2405.20159

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