Algorithm for computing Representations of the Braid Group and Temperley-Lieb algebra

TL;DR

This paper introduces an algorithm to compute and manipulate planar diagrams representing the braid group and Temperley-Lieb algebra, aiding in topological quantum computing and manifold classification.

Contribution

It presents a novel algorithm for generating and multiplying planar diagrams within the Temperley-Lieb algebra framework.

Findings

Algorithm efficiently computes all planar diagrams in a given dimension.

Enables multiplication and matrix representation of planar diagrams.

Facilitates applications in topological quantum algorithms and manifold classification.

Abstract

The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams which are Kauffman's reduction of the braid group algebra to the Temperley-Lieb algebra. We introduce an algorithm for computing all planer diagrams in a given dimension. The algorithm can also be used to multiply planer diagrams and find their matrix representation.