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

TL;DR

This paper introduces a new approach to constructing universal weight systems for knot invariants using color Lie algebras, specifically a minimal $ extbf{Z}_2^2$-graded Lie algebra, revealing relations to existing algebraic structures.

Contribution

It demonstrates how color Lie algebras can generate universal weight systems for knot invariants, including detailed analysis of a specific four-dimensional example and its relations to known algebraic weight systems.

Findings

Constructed a universal weight system from a $ extbf{Z}_2 imes extbf{Z}_2$-graded Lie algebra.

Derived relations and recurrence formulas for the weight system.

Showed the weight system as a hybrid of $sl(2)$ and $gl(1|1)$ weight systems.

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 $gl(1|1)$.