Universal weight systems from a minimal $\mathbb{Z}_2^2$-graded Lie algebra

Abstract

Color Lie algebras, which were introduced by Ree, are a graded extension of Lie (super)algebras by an abelian group. We show that the color Lie algebras can be used to construct universal weight systems for knot invariants of of Vassiliev and Kontsevich. As a simple example, we take $\mathbb{Z}_2 \times \mathbb{Z}_2$ as the grading group and consider the four-dimensional color Lie algebra called $A1_{\epsilon}$. The weight system constructed from $A1_{\epsilon}$ is studied in some detail and some relations between the weights, such as the recurrence relation for chord diagrams, are derived. These relations show that the weight system from $A1_{\epsilon}$ is a hybrid of those from $sl(2)$ and $gl(1|1)$.