  Weight systems are functions on chord diagrams satisfying so-called
Vassiliev's $4$-term relations. They are closely related to finite type knot
invariants introduced by Vassiliev.
  Certain weight systems can be derived from graph invariants. Another main
source of weight systems are Lie algebras. In recent papers, the weight systems
associated to Lie algebras ${\mathfrak gl}(N)$ were unified in a universal
${\mathfrak gl}$-weight system, which takes values in the ring of polynomials
in infinitely many variables. The unification has been achieved by extending
the ${\mathfrak gl}(N)$-weight systems from chord diagrams to arbitrary
permutations.
  A natural question then arises, namely, which already known weight systems
can be obtained from the universal ${\mathfrak gl}$ weight system. In addition
to understanding the internal relationship between weight systems, knowing that
a given weight system can be induced from the ${\mathfrak gl}$-weight system
would immediately lead to extending the former to arbitrary permutations.
  To each chord diagram, one can associate a graph, called the intersection
graph of the chord diagram. Certain weight systems are completely determined by
the intersection graphs. In general, the relationship between Lie algebra
weight systems and polynomial graph invariants looks rather complicated.
  Our main result in the present paper consists in showing that the well-known
graph and delta-matroid invariant, the interlace polynomial, can be induced
from the universal ${\mathfrak gl}$-weight system. We provide an explicit
substitution making the ${\mathfrak gl}$-weight system into the interlace
polynomial for chord diagrams and their intersection graphs.

Combinatorics