Atlas of subfactors
This is the home of the nascent "Atlas of subfactors". We're aiming to enumerate all the small rank small index subfactors by computer.
Along the same lines we'll also enumerate unitary spherical fusion category of small rank which are tensor generated by an object of small dimension.
Our project has several independent parts:
 Enumerate all possiple fusion graphs (aka principal and dualprincipal graphs) of small rank and small index.
 Implemented for subfactors, mostly implemented for selfdual objects in fusion categories
 Implement known tests in the literature which allow you to eliminate possible fusion graphs. For example:
 objects of dimension smaller than 2 are given by the Beraha numbers
 all dimensions must be algebraic integers
 the index must be a cyclotomic number
 tests from Jones' "Annular Structure of Subfactors" and "Quadratic Tangles"
 For any fusion graphs that pass all the known tests solve for all possible fusion rules
 In the subfactor case we have a working program
 Given a fusion graph find generators and relations in the bipartite graph planar algebras (see Emily's upcoming Haagerup paper)
 Incomplete, but a lot of progress has been made!
 Take all known constructions of Subfactors and find them in the lists that we have produced.
 Not started yet, but Scott's Quantum Groups package will be useful.
Contents
Getting started
The mathematica code we have thus far can be found here. You can either download individual notebooks, or, if you have subversion, obtain a local working copy with the command
svn checkout http://tqft.net/svn/fusionatlas
The notebook bigraph3.nb
is currently the best place to start; most examples below are taken from this.
Examples
Here's some examples of what we can already do.
FindAllBigraphs
The principal and dual princiapl graphs of a subfactor encode the fusion rules for the tensor generator. In the subfactor setting these are always bipartite graphs, or bigraphs.
The function FindAllBigraphs
takes two arguments; a maximum dimension (for the generating object), and a maximum rank. More documentation of the code below can be found in Entering a (graded bipartite) graph and Computing properties of graphs
Start by loading the FusionAtlas` package
In[1]:= <<FusionAtlas`
Loading FusionAtlas` version 0 Read more at http://tqft.net/wiki/Atlas_of_subfactors
Now, let's say we want all bipartite graphs having 5 or fewer vertices, with dimension less than or equal to 2. The option "Verbose" > True
explains how other bigraphs were ruled out.
In[2]:=

FindAllBigraphs[2, 5, "Verbose" > True]

Out[2]=

{Allowed bigraphs >
{GradedBigraph[{{1}}], GradedBigraph[{{1}}, {{1}}],
GradedBigraph[{{1}}, {{1}, {1}}],
GradedBigraph[{{1}}, {{1}, {1}, {1}}],
GradedBigraph[{{1}}, {{1}}, {{1}}],
GradedBigraph[{{1}}, {{1}}, {{1}}, {{1}}]},
Disallowed bigraphs >
{{GradedBigraph[{{1}}, {{1}}, {{1}, {1}}],
Bimodule dimensions less than 2 must satisfy the GHJ condition.},
{GradedBigraph[{{1}}, {{1}, {1}}, {{1, 0}}],
All bimodule dimensions must be at least 1.}}}

The first argument of FindAllBigraphs
can also be a range. Here we find all bigraphs with dimension between 2 and 2.1, with rank at most 10.
In[3]:=

FindAllBigraphs[{2, 2.1}, 10]

Out[3]=

{GradedBigraph[{{1}}, {{1}}, {{1}}, {{1}, {1}}, {{1, 0}, {1, 0}}],
GradedBigraph[{{1}}, {{1}}, {{1}}, {{1}, {1}}, {{1, 0}, {0, 1}},
{{1, 0}, {0, 1}}]}

The function DisplayBigraph
makes it easy to see which graph a sequence of matrices represents. For example, the two above are the dual Haagerup and Haagerup principal graphs:
In[5]:=

DisplayBigraph[GradedBigraph[{{1}}, {{1}}, {{1}}, {{1}, {1}}, {{1, 0}, {1, 0}}]]

Out[5]=

Graphics

In[7]:=

DisplayBigraph[GradedBigraph[{{1}}, {{1}}, {{1}}, {{1}, {1}}, {{1, 0}, {0, 1}}]]

Out[7]=

Graphics

GraphsByGlobalDimension
The function GraphsByGlobalDimension[D, k]
finds all bigraphs with global dimension at most D, with no more than k vertices with dimension 1. (All bigraphs with index < 4 are ignored.)
Todo list
 Tensorsolver
 Make the tensorsolver work for fusion graphs. (Currently it only accepts bifusion graphs.)
 Reexpansion
 Extract the fusion graph for the even part from a bifusion graph.
 Make google see this page
 Do something so that qBinomial in the KnotTheory package and the FusionAtlas package aren't in conflict
 Modify the cyclotomic test so that it only checks the square of the dimension of the generator (ie, the index)
Documentation
 Documentation for Entering a (graded bipartite) graph and Computing properties of graphs.
 Documentation for finding (bi)fusion algebras does not exist yet.
 Documentation for graph planar algebra arithmetic does not exist yet.