This is a certificate proving that the sl(2)-weight system does not extend to all graphs with 9 vertices. It consists of 3300 4T-relations and resolves into a combination of 5006 intersection graphs.

cert.9.int.csv

In cert.9.int.csv, precomputed values of the sl(2)-weight system on relevant intersection graphs are provided. The Killing form is the chosen form for the weight system. The Chmutov-Varchenko chord deletion relation is used to compute the values.

CSV fields:

• graph6: the intersection graph in the graph6 format;
• chord diagram: an arbitrary chord diagram realising this intersection graph;
• sl2: the computed value of the sl(2)-weight system.

cert.9.csv

In cert.9.csv, a combination of 4T-relations with non-zero sum and all non-zero graphs among intersection graphs is provided. Graphs are stored together with pairs of vertices (x, y). Each relation is obtained the following way:

• term 1, sign +1: the original graph;
• term 2, sign -1: the original graph where for each neighbour v of x except y, edge (y, v) is switched;
• term 3, sign +1: the original graph where for each neighbour v of x except y, edge (y, v) is switched and edge (x, y) is also switched;
• term 4, sign -1: the original graph where edge (x, y) is switched.

CSV fields:

• graph6: the initial graph in the graph6 format;
• x, y: the vertices chosen for this relation;
• coef: the coefficient associated with this relation.

check.sage

In check.sage, a SageMath program verifying the result is implemented. First it verifies the result using all the data, then it verifies that cert.9.int.csv is correct. The latter step is significantly longer and might take over an hour.

To run it, one needs to have installed:

1. SageMath.
2. Python library tqdm (used to track progress). If not available system-wide, on a Unix-like system it can be installed locally with:

python -m venv env
source ./env/bin/activate
pip install tqdm

Then the script can be executed with the sage check.sage command.

• Daniil Fomichev