Local defining relations for the cellular algebra

Establish a set of defining relations for the algebra \(\mathcal K_L\) such that every relation is supported on at most six strands, meaning that all generators appearing in a relation act nontrivially only on a common set of at most six strand positions.

Background

The paper constructs a web presentation for the algebra $\mathcal K_L$, whose basis consists of $A_2$ webs and whose local loop, digon, and square reduction rules provide a complete web-based form of the second fundamental theorem. However, an ordinary algebra presentation requires an explicit generating set of relations in the free algebra on the local generators.

The authors identify writing these defining relations explicitly as an intermediate problem beyond the web presentation. Based on their local-surgery analysis, they conjecture that all defining relations can be chosen to involve no more than six strand positions, which would establish a strong locality property for an ordinary presentation of $\mathcal K_L$.