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Description

The jellyfish algorithm is an evaluation algorithm on subfactor planar algebras originally appearing in the work of Bigelow-Morrison-Peters-Snyder in 2012. Planar algebras are a collection of vector spaces along with multilinear maps corresponding to planar tangles. Planar algebras are often defined using diagrammatic generators and relations where the kth vector space is the linear combination of planar diagrams with k strands on the top and bottom that can be formed with the generators and nonintersecting strands. For a planar algebra to be a standard invariant of a subfactor, it must have a one-dimensional 0th vector space. This is often the toughest criteria to check that a planar algebra satisfies. The jellyfish algorithm is most commonly used to bound the dimension from above. In 2025, M. showed that the jellyfish algorithm can also be used to bound the dimension from below. In this talk, we will cover the basics of planar algebras and the jellyfish algorithm, then go over ways in which to use the algorithm. That is, we will first show how to use the jellyfish algorithm to show that the dimension of the 0th vector space is at most 1, then, using results from M. 2025, show that the jellyfish algorithm can be used to show that this vector space has dimension at least 1.