Finite-type invariants of knots, links, and graphs

We define finite-type invariants for graphs as functionals on certain finite-dimensional vector spaces generated by spatial graphs. These invariants are generalizations of Vassilievʹs knot invariants to links and graphs, but our methods are quite different from those used in his paper. We show how to calculate finite-type invariants. In the case of links, we show a way to relate the invariants on links with different numbers of components, and we generalize a theorem of Birman and Lin to show that the Jones, HOMFLY, and Kauffman polynomials can be interpreted as sequences of finite-type invariants.