  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
$K_{2,n}$.
