On Spaces of connected Graphs III The Ladder Filtration

A new filtration of the spaces of tri-/univalent graphs B u m that occur in the theory of finitetype invariants of knots and 3-manifolds is introduced. Combining the results of the two preceding articles, the quotients of this filtration are modeled by spaces of graphs with two types of edges and four types of vertices, and an upper bound for dimB u m in terms of the dimensions of the filtration quotients is given. The degree m up to which B u m is known is raised by two for u = 0 and u = 2. 0 Definitions and former results This paper is based heavily on results of the preceding articles [4] and [5]. In this section we briefly repeat the relevant definitions and list results of [4] that will be used. First recall, that in the context of Vassiliev theory graphs with vertices of order one and three (uni-, trivalent vertices) are subjected to some local relations (i.e. a quotient of a vector space with a basis of such graphs is quotiented by linear combinations of graphs differing only inside a small region in way that is specified by the ”relation”. Edges that go out off the region are clipped, but the resulting ”ends” should not be confused with univalent vertices: all relations in this paper do not involve any univalent vertices. Furthermore, the ends are supposed to be arranged correspondingly in all terms of a relations). The incoming edges at every trivalent vertex are supposed to be cyclically ordered and the graphs are always drawn such that these orderings correspond with the orientation of the plane. A connected graph with 2m vertices will be called diagram of degree m. Definition 0.1 B u m := Q〈 diagrams of degree m with u univalent vertices 〉 / (AS), (IHX) B̂ u m := Q〈 diagrams of degree m with u univalent vertices 〉 / (AS), (IHX), (x) where (AS), (IHX) and (x) are the following local relations ((x) is rather a family of relations):