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# 1. Introduction

In the late 1980s, Vassiliev [Va1,Va2] suggested to study knot invariants by studying the space of all knots. The first new ingredient thus added is the notion of neighboring'' knots, knots that differ in only one crossing, and the idea that one should study how knot invariants change in the neighborhood'' of a given knot. This idea led to a definition of a certain class of knot invariants, now known as Vassiliev'' or finite type'' invariants. By now more than 350 papers have been written on the subject; see [B-N2]. Vassiliev's original definition (independently discovered by Goussarov [Go1,Go2] at roughly the same time) was formalized in a different way by Birman and Lin [BL], and later re-interpreted by analogy with multi-variable calculus by Bar-Natan [B-N1].

The basic idea in Bar-Natan [B-N1] was that differences are cousins'' of derivatives, and hence one should think of the difference between the values of a knot invariant V on two neighboring knots as a derivative'' of the original invariant. Repeating this, we find that iterated differences (as considered by [Va1,Va2,Go1,Go2,BL,B-N1]) of values of V in the neighborhood of some knot should be thought of as multiple derivatives. A Vassiliev'' or finite type'' invariant of type m is then the analog of a polynomial -- an invariant whose m+1st derivatives, or m+1st iterated differences, vanishes.

Clearly, these ideas are very general, and knots (and even topology in general) are just a particular case. Whenever an appropriate notion of neighborhood of an object'' exists, one can talk about finite type invariants of such objects. This leads to many different species'' of finite type invariants. Let us mention just a few:

• The usual notion of nearness of knots, knots that differ at only a single crossing, leads to the usual Vassiliev invariants.
• Similarly, one can define Vassiliev'' invariants of braids, links, tangles, knotted graphs, etc.
• Goussarov [Go3] has also an alternative notion of a neighborhood of a knot (or link), defined by interdependent modifications''. This notion leads to a different (though at the end, equivalent) theory of finite type invariants of knots and links.
• Two algebraically split links (links whose linking numbers all vanish) can be considered neighboring if they differ by the simultaneous flip of two opposite crossings between two given components (such a double flip preserves linking numbers, whereas a single flip doesn't). See Figure 1. This leads to a little known but probably interesting theory of finite type invariants of algebraically split links.

• Several authors [] have considered several notions of finite type invariants of plane curves.
• Ohtsuki [Oh1] considers two integral homology spheres to be neighboring (roughly) if they differ by a single surgery. This leads to a notion of finite type invariants of integral homology spheres. Several variants of his definition were considered in [Ga1,GL1,GO1,GL2,GO2,Ga2,GL3]

A finite type theory automatically comes bundled with several spaces that play a significant role in it. In the best known case of knots, these are the spaces of chord diagrams, 4T relations, weight systems, chord diagrams modulo 4T relations, relations between 4T relations, and a few lesser known spaces that should probably be better known. There is a general theory of finite type invariants'', defined in terms of these spaces, in which one attempts to classify finite type invariants by first classifying their potential mth derivatives, and then by studying which of those potential derivatives can actually be integrated to an honest invariant. I should say that though this general theory'' is rather small, it is also rather interesting (with the most interesting parts developed by M. Hutchings [Hu] and private communication), and insufficiently well known even in the case of the usual finite type invariants of knots.

The purpose of this paper is twofold:

1.
To state (and propagate) this general theory of finite type invariants of anything. Namely, to construct, name, and study the relationships between those spaces that come automatically with every finite type theory, especially from the perspective of the integration theory of weight systems''. We first do it in Section 1 on a well known example, the original finite type theory of knots. We then extract some general features from this example and give them general names; this is done in the rather short Section 2.

2.
To list many of the currently known finite type theories, and figure out (to the degree that is now possible) what these associated spaces are on a species by species basis. Our list takes the form of a classification''1. The top subdivision is into the classes of Knotted Objects'', 3-Manifolds'' and Plane Curves'', and these classes are described in Sections 45 and 6, respectively.

Acknowledgement: I wish to thank Mike Hutchings for the inspiration to write this paper and for telling me his ideas about integration theory. I also wish to thank E. Appleboim and H. Scolnicov for many ideas and conversations.

#### Footnotes

... classification''1
classification (klàs´e-f-k´shen), in biology, the systematic categorization of organisms. One aim of modern classification, or systematics, is to show the evolutionary relationships among organisms. The broadest division of organisms is into kingdoms, traditionally two-Animalia (animals) and Plantae (plants). Widely accepted today are three additional kingdoms: the Protista, comprising protozoans and some unicellular algae; the Monera, bacteria and blue-green algae; and the Fungi. From most to least inclusive, kingdoms are divided into the following categories: phylum (usually called division in botany), class, order, family, genus, and species. The species, the fundamental unit of classification, consists of populations of genetically similar, interbreeding or potentially interbreeding individuals that share the same gene pool (collection of inherited characteristics whose combination is unique to the species). Copyright ©1995 by Columbia University Press. All rights reserved.

Next: 2. Background Material Up: Finite Type Invariants by Previous: Contents
Dror Bar-Natan
2000-03-19