We show that the adjacency matrices of the intersection graphs of chord
diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and
hence give rise to weight systems. Among these weight systems are those
associated with the Conway and HOMFLYPT polynomials. We extend these ideas to
looking at a space of {\it marked} chord diagrams modulo an extended set of
2-term relations, define a set of generators for this space, and again derive
weight systems from the adjacency matrices of the (marked) intersection graphs.
Among these weight systems are those associated with the Kauffman polynomial.Comment: 20 pages. This version has been substantially revised. The results
are largely the same, but the proofs have been reconceptualized in terms of
various 2-term relations on chord diagrams and graph
