# New hyperfinite subfactors with infinite depth

## TL;DR

This paper constructs new hyperfinite subfactors with TLJ standard invariant at all indices between 4 and 5, including the first example with index 5, revealing surprising structures in subfactor theory.

## Contribution

It introduces explicit constructions of hyperfinite subfactors with TLJ invariants at indices between 4 and 5, including the first with index 5, expanding understanding of subfactor indices.

## Key findings

- Existence of hyperfinite subfactors with TLJ invariant at all indices between 4 and 5.
- First example of hyperfinite subfactor with trivial invariant and index 5.
- Multiple hyperfinite subfactors with index 3 + √5, showing rich structure.

## Abstract

We construct new hyperfinite subfactors with Temperley-Lieb-Jones (TLJ) standard invariant and Jones indices between $4$ and $3 + \sqrt{5}$. Our subfactors occur at all indices between $4$ and $5$ at which finite depth, hyperfinite subfactors exist. The presence of hyperfinite subfactors with TLJ standard invariant, and hence of infinite depth, at these special indices comes as a surprise. In particular, we obtain the first example of a hyperfinite subfactor with ``trivial'' (i.e. TLJ) standard invariant and integer index $>4$, namely with index $5$. This is achieved by constructing explicit families of nondegenerate commuting squares of multi-matrix algebras based on exotic inclusion graphs with appropriate norms. We employ our graph planar algebra embedding theorem [BC25], a result of Kawahigashi [Kaw23] and other techniques to determine that the resulting commuting square subfactors have indeed TLJ standard invariant. We also show the existence of a hyperfinite subfactor with trivial standard invariant and index $3 + \sqrt{5}$. It is interesting that this index features at least two distinct infinite depth, irreducible hyperfinite subfactors and several finite depth ones as well. Our work leads to the conjecture that every Jones index $>4$ of an irreducible, hyperfinite (finite depth) subfactor can be realized by one with TLJ standard invariant.