On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite
Zhuoke Yang
TL;DR
The paper computes the $\mathfrak{sl}_3$ weight system on chord diagrams with intersection graphs $K_{2,n}$ within the Vassiliev framework, revealing a dependence on the second Casimir $c_2$ (and $c_3$ where applicable) after projecting to primitives. It develops a Hopf-algebraic and Jacobi-diagram approach, leveraging the leaf lemma and STU relations to derive a recurrence for a family of Jacobi diagrams $J_{i,j}$ and to obtain explicit closed forms and generating functions for $K_{2,n}$. A key outcome is that primitive projections of these diagrams can be expressed as combinations of connected Jacobi diagrams with at most 4 legs, leading to a conjecture that generalizes Lando’s observations for $\mathfrak{sl}_2$ to arbitrary $\mathfrak{g}$. The work connects to Filippova’s results on $\mathfrak{sl}_2$ and motivates broader bounds on the degree of weight-system evaluations in primitive components, enriching our understanding of higher-rank Lie-algebra weight systems in the chord-diagram setting.
Abstract
Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord diagram encodes the order of double points along a singular knot. A Vassiliev invariant of order $n$ gives rise to a function on chord diagrams with $n$ chords. Such a function should satisfy some conditions in order to come from a Vassiliev invariant. A weight system is a function on chord diagrams that satisfies so-called 4-term relations. Given a Lie algebra $\mathfrak{g}$ equipped with a non-degenerate invariant bilinear form, one can construct a weight system with values in the center of the universal enveloping algebra $U(\mathfrak{g})$. In this paper, we calculate $\mathfrak{sl}_3$ weight system for chord diagram whose intersection graph is complete bipartite graph $K_{2,n}$.