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

TL;DR

This paper introduces a polynomial-time algorithm for multiplying elements in the skein algebra of a one-hole torus, simplifying computations in a complex topological quantum field theory context.

Contribution

It presents the first polynomial algorithm for skein algebra multiplication on a one-hole torus, advancing computational methods in topological quantum algebra.

Findings

Polynomial algorithm for skein multiplication

Closed-form formulas for low crossing number curves

Enhanced computational efficiency in skein algebra

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.