Jellyfish Algorithm in Subfactor Planar Algebras

• Jellyfish algorithm is a finite diagrammatic evaluation method in subfactor planar algebras, employing specialized jellyfish generators and train diagrams to systematically reduce complex diagrams.
• The method ensures principal graph stability by demonstrating that train spanning properties lead to finite spoke graphs with A-type tail extensions beyond a critical depth.
• Practical construction involves quadratic tangles and automated linear algebra, offering an explicit computational framework for verifying subfactor classification and standard invariant properties.

The jellyfish algorithm is a finite, diagrammatic evaluation and reduction procedure in subfactor planar algebras, designed specifically to exploit the algebraic and combinatorial structure imposed by certain classes of generators—called jellyfish generators—whose presence is intimately connected to the stability of the principal graph and its “spoke graph” shape. This method allows one to systematically rewrite any diagram as a linear combination of so-called train diagrams, providing both explicit computational techniques for evaluations and a structural framework for the classification of subfactors whose standard invariants have highly constrained principal graphs.

1. Jellyfish Generators and the Structure of Trains

At the core of the jellyfish algorithm are special sets of generators in the box spaces $P_{n,+}$ and $P_{n,-}$ of a planar algebra, termed jellyfish generators. Two principal archetypes are considered:

• 2-strand jellyfish generators: A set $\mathcal{S}_+ \subset P_{n,+}$ is a set of 2-strand jellyfish generators if (i) every diagram can be written as a linear combination of trains—diagrams formed by concatenations of elements from $\mathcal{S}_+$ (with intervening Temperley–Lieb pieces)—and (ii) their span (together with the Jones–Wenzl idempotent $f^{(n)}$) is closed under the box-space multiplication. For all $S \in \mathcal{S}_+$:

$$j^2(S) \in \operatorname{span}(\mathrm{trains}_{n+2}(\mathcal{S}_+))$$

• 1-strand jellyfish generators: Sets $\mathcal{S}_+ \subset P_{n,+}$ and $\mathcal{S}_- \subset P_{n,-}$ are involved, with relations of the form:

$$j(S) \in \operatorname{span}(\mathrm{trains}_{n+1,\mp}(\mathcal{S}_{\mp}))$$

for each $S$ in $\mathcal{S}_+$ and vice versa.

Trains in this context are diagrams with sequences of generators (aligned along their designated intervals) separated by Temperley–Lieb elements. These trains form a spanning set, ensuring every diagram in the planar algebra is expressible via these structured configurations.

2. Principal Graph Stability and the Mechanism of the Jellyfish Algorithm

There exists a fundamental equivalence between the train spanning property (i.e., the ability to reduce any diagram to a train) and the stability of the subfactor’s principal graphs beyond a certain depth, echoing results first seen in Popa’s work on principal graph stability. Formally, stability at depth $n$ holds if:

$$P_{n+1,\pm} = P_{n,\pm} + P_{n,\pm}e_{n,\pm}P_{n,\pm}$$

where $e_{n,\pm}$ are the Jones projections. The following are equivalent:

1. For all $k \geq n$, $P_{k,\pm}$ is generated by $P_{n,\pm}$ and Temperley–Lieb elements.
2. Trains from $P_{n,\pm}$ span $P_{k,\pm}$.
3. The principal graphs $\Gamma_+, \Gamma_-$ are stable for all depths $k \geq n$.

Crucially, successful application of the jellyfish algorithm assures principal graph rigidity: once spanning by trains is achieved, further "growth" of the graph involves only the addition of $A$-type tails.

3. Spoke Graphs and the Existence of Jellyfish Generators

A spoke graph, as defined in the literature, is a tree that is essentially linear except for a unique central vertex of higher valence. Specifically, a simply laced spoke graph is a bipartite tree with two distinguished vertices—a basepoint (a leaf) and a central vertex $c$—where only $c$ may have valence exceeding two. Spoke graphs with multiple edges connecting to the central vertex (excluding the basepoint path) are also considered.

A main result is the equivalence:

• Existence of 1-strand jellyfish generators in $P_{n,+} \cup P_{n,-}$ $\Longleftrightarrow$ $\Gamma_+$ and $\Gamma_-$ are finite spoke graphs.
• Existence of 2-strand jellyfish generators in $P_{n,+}$ $\Longleftrightarrow$ $\Gamma_+$ is a finite spoke graph and $\Gamma_-$ is stable at depths $\geq n+1$.

Thus, the spoke graph condition offers a precise characterization: only those principal graphs with a unique branching (at the central vertex) and otherwise $A$-type tails admit a jellyfish algorithm-based evaluation and classification scheme.

4. Methodology: Quadratic Tangles and Automated Construction

Construction of subfactors with the desired principal graphs utilizes explicit generators and the systematic derivation of jellyfish relations, leveraging Jones’ quadratic tangle formulas. The workflow is as follows:

1. Select generators (typically two, labeled $A, B$) in the depth-$n$ box space.
2. Compute quadratic tangles (e.g., $A \circ A$, $A \circ B$, etc.).
3. Calculate their inner products modulo annular consequences to identify a basis for QTAC (quadratic tangles in annular consequences).
4. Express elements using a natural annular basis, leading to equations of the form:

$f(2n+2) \circ j(S) = \sum_i Y_{S,i} U_i(S)$

where $j(S)$ is a box jellyfish diagram and $U_i(S)$ are basis annular consequences.

1. Assemble the relations into a "jellyfish matrix" $J$ and (if of full rank) compute its left inverse using:

$J_L = (J^* J)^{-1} J^*$

1. Derive explicit box jellyfish relations, which serve as the reduction rules.

This entire procedure is reducible to linear algebra over suitable number fields and is amenable to automation. Notably, the construction of the 4442 (a novel example) as well as the previously known 3333, 3311, and 2221 spoke subfactors were carried out in this manner, using tools such as Mathematica notebooks and the FusionAtlas software suite.

5. Two-Strand Jellyfish Relations and Annular Consequences

The extension to two-strand jellyfish relations involves a more intricate combinatorial and diagrammatic analysis, focused on the computation of the space of second annular consequences. For a generator $S$, the key relation takes the schematic form:

$$f(2n+4) \circ j_2(S) = JL K \left( \sum_{P,Q} f(P \circ Q) \right)$$

where $J$ is the jellyfish matrix, $L = (J^*J)^{-1}J^*$ its left inverse, and $K$ corresponds to the reduced trains. This procedure analytically mirrors Jones’ approach to quadratic tangles, adapted to the combinatorics of the two-strand setting, and is essential for the construction and verification of subfactors where only one principal graph is a spoke graph.

Work in this area also provides a systematic paper of the annular bases and their duals, crucial for reformulating diagrammatic relations and ensuring that all reduction steps are valid modulo Temperley–Lieb elements and annular consequences.

6. Applications and Significance

Once the jellyfish relations are established for a given set of generators, every diagram in the planar algebra can be reduced inductively to a linear combination of trains. This provides:

• An explicit, algorithmic pathway for evaluation of closed diagrams in the zero-box space.
• An effective means to verify the structure and evaluability of the planar algebra generated by the candidate generators.
• A direct method of verifying that constructed subfactors possess the intended principal graphs (e.g., verifying supertransitivity and the shape of 4442 in practice).
• Systematic generation and classification of subfactors with small index and specifically constrained combinatorics, as demonstrated in constructions of new subfactors such as 4442.

Moreover, the methodology generalizes—using higher annular consequences and train-basis computations—allowing for the classification of broader families of planar algebras within the constraints set by the spoke graph shape and stability.

7. Broader Context and Theoretical Connections

The jellyfish algorithm, through its interplay with principal graph stability and explicit diagrammatic reduction, anchors the classification of subfactor planar algebras in a strong combinatorial and algebraic framework. Its development builds upon foundational results by Popa, Jones, and subsequent systematic automation by researchers such as Bigelow, Penneys, Morrison, and collaborators. The algorithm not only enables construction and recognition of specific subfactors but also elucidates the inherent rigidity and structure of the standard invariant when principal graphs satisfy spoke or stability conditions.

The approach features a recursive reduction mechanism, reliance on explicit basis computations in annular categories, and admits automation—a feature crucial for handling increasingly complex examples or number-theoretic constraints. Its connection to explicit algebra presentations and the systematic calculation of algebraic compressions further cements its role in advancing the understanding and expansion of subfactor theory.