  A weight system is a function on chord diagrams that satisfies the so-called
four-term relations. Vassiliev's theory of finite-order knot invariants
describes these invariants in terms of weight systems. In particular, there is
a weight system corresponding to the colored Jones polynomial. This weight
system can be easily defined in terms of the Lie algebra $\mathfrak{sl}_2$, but
this definition is too cumbersome from the computational point of view, so that
the values of this weight system are known only for some limited classes of
chord diagrams.
  In the present paper we give a formula for the values of the
$\mathfrak{sl}_2$ weight system for a class of chord diagrams whose
intersection graphs are complete bipartite graphs with no more than three
vertices in one of the parts.
  Our main computational tool is the Chmutov--Varchenko reccurence relation.
Furthermore, complete bipartite graphs with no more than three vertices in one
of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we
deduce formulas for the projection onto the subspace of primitive elements
along the subspace of decomposable elements in these subalgebras. We compute
the values of the $\mathfrak{sl}_2$ weight system for the projections of chord
diagrams with such intersection graphs. Our results confirm certain conjectures
due to S.K.Lando on the values of the weight system $\mathfrak{sl}_2$ at the
projections of chord diagrams on the space of primitive elements.
