Incorporating trivalent vertices into the N-graph defect framework

Develop a defect-category framework that accommodates N-graphs with both hexagonal and trivalent vertices, extending the current construction based on the symmetric-group presentation so that the resulting basic-gons and defect data consistently encode trivalent vertices as well.

Background

The paper realizes hexagonal-vertex N-graphs via the category Bord_2^{def,cw}(𝓟_{S_n}) using the standard presentation of the symmetric group. However, general N-graphs can also feature trivalent vertices, which are not yet captured in the current setup.

The authors highlight the gap and suggest that a larger defect category may be needed, possibly relating to cluster or Soergel-bimodule structures, to uniformly treat both hexagonal and trivalent N-graph vertices.