Integral Expressions for the Vassiliev Knot Invariants
(Figures redrawn January 28, 1999)
Abstract
It has been folklore for several years in the knot theory community that certain integrals on configuration space, originally motivated by perturbation theory for the ChernSimons field theory, converge and yield knot invariants. This was proposed independently by Gaudagnini, Martellini, and Mintchev [11] and BarNatan [4]. The analytic difficulties involved in proving convergence and invariance were reportedly worked out by BarNatan [4, 3], Kontsevich [13, 16], and Axelrod and Singer [1, 2].
But I know of no elementary exposition of this fact. BarNatan [3] only proves invariance for the degree 2 invariant. Kontsevich’s exposition is decidedly nonelementary and leaves many details implicit. Axelrod and Singer’s papers take a physics point of view (and are thus difficult for mathematicians to read) and only discuss the related invariants for 3manifolds. More recently, Bott and Taubes [7] have explained the degree 2 invariant from a purely topological point of view, but again omit the higher degree cases.
This thesis is an attempt to remedy this lack. I adopt an almost exclusively topological point of view, rarely mentioning ChernSimons theory. For an explanation of the physics involved, see, for instance, BarNatan [3].
There are also a few new results in this thesis. These include a new construction of the functorial compactification of configuration space (Section 3.2) as well as some variations on the integrals. For a suitable choice of this variation, the integral reduces to counting tinkertoy diagrams (Section 4.5). In particular, the invariants constructed take values in .
Contents
Chapter 1 Introduction
1.1 Motivation
First, I’ll present some motivation for introducing these invariants, primarily intended for topologists.
A (mathematical) knot^{1}^{1}1As opposed to a physical knot, which has a finite thickness and has ends. is a loop in : an embedding . An embedding is a smooth map with two conditions:

Immersion condition: ,

Non selfintersecting: ,
The first condition is purely local and not terribly interesting. (In fact, removing that condition yields an essentially equivalent space.) The second condition is much more interesting. It’s a global condition and is what makes the study of knots much harder than the study of loops. How can we reduce it to a finite condition? One answer comes from the tradition of raising maps to the product of the spaces involved. Observe that from the map , we can construct
(1.1) 
This may seem like a trivial remark, but it has farreaching consequences. In particular, because of condition 2, restricts to a map between the configuration spaces. The configuration space of points in a manifold is the space of distinct points in . Formally,
(1.2) 
and, if is an embedding, we can define
(1.3) 
as the restriction of to . You can therefore try to gain information about the map by looking at the map on the configuration spaces .
In this thesis, I begin to carry out this program in the case of knots. has a useful set of homology generators given by the pullbacks of a 2form on via the direction map for each pair of points in . By pulling back these forms to and taking sums so that the boundary terms cancel, I will construct explicit formulas for the Vassiliev knot invariants.
1.2 Linking Number
To help explain these ideas, I’ll give a simple example. The linking number is a wellknown invariant of twocomponent links. (An component link is an embedding of a disjoint union of circles in .) Label the two circles and and consider the set of crossings—points of that lie directly above a point of . If the link is in general position, there will only be a finite number of crossings, and they will project to distinct points in the plane. Count up the crossings with signs, as in Figure 1.1. A crossing is counted with a positive sign if runs from left to right as you look along the direction of , negative otherwise. See Figure 1.2 for an example of this computation of the linking number.
Because of the sign convention, the linking number turns out to be an invariant of the link; it doesn’t depend on the position of the link in space. See Figure 1.3, which shows invariance under the one Reidemeister move (see Kauffman [12]) in which the set of crossings changes.
Configuration space provides a nice way to understand this combinatorial formula topologically. Because of the type of crossings that we count, it makes sense to consider the subset of in which one point lies on and the other on . Since , this is isomorphic to , which is a torus.
As in Section 1.1, we get a map . There is also a map^{2}^{2}2In fact, a homotopy equivalence. , given by
(1.4) 
(the denominator does not vanish in ). Composing these two, we get a map from the torus to the sphere. The “crossings” we counted above were just those points on the torus mapping to the vertical direction on the sphere. If we orient both the torus and the sphere, the sign of the crossing defined above turns out to be the sign of the local mapping from the torus to the sphere (i.e., whether it locally preserves or reverses orientation), which will be nondegenerate for knots in general position. See Figure 1.4.
In fact, this number is independent of the direction ( point on the sphere) that we choose, as long as it’s a regular value. See Figure 1.5.
This type of counting is wellknown in topology; it is known as the degree, and can be defined for any map between oriented compact manifolds of the same dimension .
To define the degree rigourously, note that the highest cohomology of a compact oriented manifold , , is always 1dimensional and has a natural generator (so that the integral of over the manifold is 1); this is called the fundamental class of . The map induces a map , taking the fundamental class of to some multiple of the fundamental class of , which can be computed by integrating over . This multiple turns out to be exactly the degree. By Sard’s theorem, we know that there is some regular point of , i.e., points for which is a diffeomorphism in the neighborhood of each inverse image of . Since is compact, there are only a finite number of inverse images of . If we take a representative of the fundamental class of that is sufficiently localized around , then the integral of over is exactly the degree as defined above.
But you need not pick a localized . In our case, of a map from , it’s natural to choose to be the unique SO(3) invariant volume form on ,
(1.5) 
(using physicists’ conventions: the components of the 3vector are , and repeated indices in an expression are to be summed over, while is the basic totally antisymmetric tensor), then the integral of the pullback of over the knot becomes becomes
Linking number  (1.6)  
(1.7) 
The first expression is again written in physicists’ notation (the are the components of the embedding of the ’th circle into ). In the second, I use notation from Bott and Taubes [7]: .
This is the simplest case of the integrals considered in this thesis. (Although I won’t consider links any further, the generalization of these integrals to links is straightforward.)
1.3 Back to Knots
But this case is misleadingly simple. Consider trying to modify the linking invariant to get an invariant of a single knot. As before, there is a canonical map . But now is not compact (it is the torus minus the diagonal) so there is no welldefined degree. Alternatively, if you compactify by adding a boundary, the number of inverse images of a point in changes as you cross the image of the boundary. See Figure 1.6.
The resolution of this difficulty is to take linear combinations of maps from spaces associated with the knot, each with a boundary, in such a way that the boundaries exactly cancel out in the target manifold. The resulting “degree” will automatically be an invariant of knots.
In the case of , the necessary correction term comes from a framing of the knot: a nowhere zero section of the normal bundle to the knot (as imbedded in ). (A framed knot can also be thought of as an imbedded (oriented or twosided) ribbon in .) With this correction, you get the framing number of the knot: the linking number of the knot with a copy slightly displaced in the direction of the framing.
But it would take us too far afield to describe this in more detail. Instead, let’s go to level 4, where there is an honest knot invariant (rather than an invariant of framed knots). That is, consider . Unlike , this is a disconnected space. Pick the connected component in which the points 1, 2, 3, 4 appear in that (cyclic) order around the circle. There are now maps (for , ), given by normalizing the vector . To define a degree of any sort, we’ll need a map from the 4dimensional space to another 4dimensional manifold. We can find one such by considering, for instance, . (This is really the only possible choice using our . All of 1,2,3,4 must be represented, or the map is degenerate, and if we have a term like (with two adjacent points on the circle) we’ll get a boundary just like that from on , which requires a framingdependent correction.)
The term to counter the boundary of the map turns out to be slightly more complicated. It involves a map from , the space of 4 points in , with the first 3 restricted to lie on the knot . From this 6dimensional space we consider the map . Then the resulting integral
(1.8) 
converges and is a knot invariant. (As before, , with the normalized volume form on the sphere.)
Raoul Bott and Cliff Taubes [7] proved this integral is invariant. (Earlier, Dror BarNatan [4] had proven invariance in another way.) Their proof relies crucially on the functorial compactification of the configuration space , described in detail in Section 3. This compactification yields a manifold with corners, whose (codimension 1) faces correspond to the points in a subset of size approaching each other. (Since the compactification is functorial, the same description holds for the other spaces involved, and .)
The proof is essentially Stokes’ theorem: the variation of the integral of a (closed) form is the integral of the form on the boundary. (This yields a 1form on the space of knots .) See Proposition 4.2 for the precise statement. Bott and Taubes showed that (a) either integral restricted to any face in which more than two points come together (as a 1form on the space of knots) vanishes and (b) the integrals restricted to the principal faces, those in which exactly two points come together, cancel each other out.
The principal faces are more interesting, so let’s look at them first. On the face of in which points 1 and 2 come together, the form becomes . (The notation indicates that is equal to . It also has another meaning, relating to the coordinates on : on this face, coordinates are the positions , , and . Thus this form can be considered a form on .) See Section 3.1.) Similarly for the other principal faces (with 2 and 3, 3 and 4, and 4 and 1 colliding) of . In , consider the face in which points 1 and 4 come together. On this face, the form becomes . But, on this face, the map varies over without changing the directions to any of the other points; so we can integrate over this , yielding the form on . Similarly for the faces with 2 and 4 and 3 and 4 colliding. On the faces in which two points on the knot collide (for instance, 1 and 2), the restriction of this form vanishes.
Thus each of the four principal faces of contributes a form like . All of these yield the same 1form on . Each of the three interesting principal faces of yields another form like this, again yielding the same 1form on . Because of the factors and signs in the expression 1.8, all these forms cancel.
This result was not news to the knot theory community. This compactification of configuration space had been known and used, for instance by Axelrod and Singer [1, 2], who proved existence and invariance of the similar integral invariants for homology 3spheres, and it was common knowledge that the techniques worked for the knot invariants too. But I know of no elementary exposition. The BottTaubes paper [7] is a first step; unfortunately, it omits description of the higher invariants, which can be treated with the same methods. This thesis attempts to rectify the situation.
The invariants are all constructed from configuration space integrals of this basic type. Namely, given a graph with a cycle representing the knot, the vertices of define a configuration space , and the edges of define a map to (after choosing an ordering of the edges and a direction for each edge). The corresponding integral is the integral over of the pullback of the canonical form on (which is just the product of the canonical forms on the factors). Note that for this to be welldefined, there must be an orientation on . An ordering of the vertices of gives such an orientation.
Chapter 2 defines the Vassiliev invariants and introduces weight diagrams, with a plausibility argument why they should yield invariant integrals. Chapter 3 introduces the functorial compactification of configuration space that is essential to removing the analytic difficulties involved. Chapter 4 ties up loose ends, completing the description of the invariants, showing that the principle faces cancel, and describing the tinkertoy diagram formula. Chapter 5 shows that the contributions to the variation of the integral from most hidden or nonprinciple faces vanish. The remaining faces (the anomalous faces) are the subject of Chapter 6.
1.4 Acknowledgements
I would like to thank Ezra Getzler, Michael Hutchings, Jose Labastida, Justin Roberts, James Stasheff, Clifford Taubes, and Zhenghan Wang for helpful conversations. Dror BarNatan has helped straighten out my confusions in many ways. And Raoul Bott, my advisor, has been tremendously helpful throughout countless hours of discussion.
Chapter 2 Vassiliev Invariants and Weight Diagrams
This chapter is a brief introduction to Vassiliev invariants and the combinatorics of weight diagrams, which are basic to the construction of configuration space integrals. The reader is referred to BarNatan’s excellent paper [6] for a more indepth treatment.
Any knot invariant can be extended to an invariant of immersed circles with exactly transversal selfintersections inductively, using
(2.1)  
(2.2) 
Equation 2.2 is to be interpreted a local statement. The loose ends are connected by some paths, possibly with crossings. Note that this definition depends on the orientation on , but not the plane projection or the orientation on . This is a good definition, since (expanding out) it expresses on a “knot” with transversal selfintersections as a signed sum of the values of on the ways of resolving the singularity, which won’t depend on the order of expansion.
Definition 2.1 (The BirmanLin Condition).
One way to think of these Vassiliev invariants is as invariants of polynomial type. Taking the difference of the value of the knot invariant between the two sides of a crossing can be seen as “differentiation” of a sort. A polynomial is a function that becomes 0 if you differentiate it enough, similar to the definition of a Vassiliev invariant above.
Directly from the definition, we can see that the invariants we’ll construct will be Vassiliev invariants.
Proposition 2.2.
A sum of canonical configuration space integrals of degree (involving a configuration space with points) that is a knot invariant is a Vassiliev invariant of degree .
The proof is in Section 4.1. The essential idea is that there can only be a difference between an undercrossing and an overcrossing if there is at least one point on each strand very near the crossing and a propogator connecting them.
If you differentiate a Vassiliev invariant of degree just times, you get the analog of a constant. This is an invariant of knots with double points that doesn’t change as the knot passes through itself: this is exactly the condition that it vanishes on knots with double points. Thus the invariant evaluated on a knot with double points doesn’t depend on the embedding in space, but only on the topological structure of the image. This is completely determined by labelling the points of selfintersection of the knot and recording the labels of the points as you travel around the circle in the increasing coordinate. You can get a graphical representation (called a chord diagram) by drawing the circle as a circle and connecting points on the circle that get mapped to the same point in , as in Figure 2.1.
Each Vassiliev invariant of degree therefore determines a function on chord diagrams with chords. We can evaluate this function explicitly for configuration space integrals. By the proof of Proposition 2.2, the only configuration space integrals of degree that don’t vanish when evaluated on a knot with double points are those that involve points, all on the knot, connected by propogators in pairs that are near the double points. In particular, the graph of the propogators must be isomorphic the the graph of the chord diagram. Since the integral of a basic propogator near a double point is , the value of the configuration space integral on this knot with double points will be (up to sign) the number of such isomorphisms. See Section 4.1 for more details.
Not all functions on chord diagrams can be achieved as derivatives of knot invariants. In particular, imagine evaluating the invariant on a knot with double points and moving one strand in a small circle around one of the other double points, as on the left in Figure 2.2.
The difference as you pass through each strand is the value of the invariant on a certain chord diagram. Since you end up at the same knot you started at, the sum of the value on the four diagrams produced must be 0. Unfolding the relation, you get the sum of diagrams on the right in Figure 2.2, call the four term or relation. Functions on chord diagrams satisfying the relation are called weight diagrams. One of the consequences of this thesis is that any weight diagram can be “integrated” to yield an invariant of knots. This was proved earlier by Kontsevich using a different type of integral. See [6] for a description.
The number of weight diagrams of a given degree is an important open question. BarNatan [6] has done extensive computer computations, computing all weight diagrams up to degree 9. There seem to be a fair number of them, probably growing exponentially, although the difficulty of computation grows as a considerably faster exponential.
It’s an amazing fact that any weight diagram can be extended to a function on diagrams with internal vertices of a certain type. Namely, a trivalent graph (termed a Chinese Character Diagram by BarNatan [6]) is a connected graph with a directed cycle (with distinct vertices) and all vertices trivalent. Vertices that do not meet are the internal (or free or interaction) vertices. The value of a weight diagram on a trivalent graph will depend on an orientation on the edges incident to each internal vertex. Since the vertices are trivalent, this is a cylic order.
Proposition 2.3 (BarNatan [6],Theorem 6).
A weight diagram extends uniquely to a function on trivalent graphs with an orientation on the edges incident to each vertex satisfying the relation
(2.3) 
Proof.
It’s clear that (since trivalent graphs are connected) the relation defines inductively the value of on any trivalent graph, as long as the definition does not introduce inconsistency.
The relation is a difference of two relations involving graphs with zero or one internal vertices. Because of this, the value of on a graph with exactly one internal vertex is welldefined. The proof then proceeds by induction. Suppose has been defined on graphs with or fewer internal vertices. Consider for a graph with vertices. This can be defined from the value of on graphs with internal vertices via on any of the edges that connects an internal vertex of to the circle. If two of these edges connect to different internal vertices, then the corresponding two potential expressions for can be expanded into a sum of for graphs with vertices via at the next lower level on the other edge. The resulting graphs are the same, so the two potential expressions for are consistent. If two edges of connect the same internal vertex to the circle but there is a third edge connecting a different internal vertex to the circle, we can apply this procedure in two steps to deduce consistency. There is one remaing case, as in Figure 2.3. Since the value of the weight diagram on trivalent diagrams like this will turn out to be irrelevant, I defer to BarNatan [6] for the completion of the proof. ∎
Internal vertices arise naturally in the Feynmann diagrams for ChernSimons perturbation theory, where they represent the interactions of the theory. But they are still a mystery from other points of view. has not yet been given a sensible interpretation in terms of knots when has internal vertices. Bott and Taubes [7] have suggested that, from the differential point of view they are related to Sullivan’s minimal model for the rational homotopy of , but this has not yet been explained clearly. In any case, we have the following theorem. is, as in the introduction, the configuration space integral corresponding to the trivalent graph (with a sign yet to be defined).
Theorem.
For any weight diagram , the following sum of integrals is a knot invariant:
(2.4) 
The expression 2.4, without the correction term, is exactly the integral that would be written down in perturbation theory, but it has been decoupled from all physics considerations here.
The proof is spread out over the remainder of the paper. The basic idea is that the differential form extends smoothly to the compactification of configuration space. In this context, the variation of the integral as the knot varies can be written (up to sign) as the sum of the restrictions of the integral to the faces of configuration space. (See Proposition 4.2.)
The most interesting faces are the principle faces, which correspond to exactly two points (connected by either a segment of the knot or a propogator) approaching each other. The relation expresses the cancellation of faces in which two vertices on the knot or a vertex on the knot and an internal vertex approach each other. The remaining principle faces (two internal vertices colliding) cancel by the following proposition:
Proposition 2.4 (BarNatan [6], Theorem 6).
Every weight diagram extended to graphs satisfies the relation:
(2.5) 
I will not repeat the proof here. It reduces to the case when one of the edges is connected to the circle, and in this case you get a sum of terms very much like the Jacobi identity for matrices.
The factor , the size of the automorphism group of the graph , arises for the same reason a similar factor appeared for the evaluation of an integral on a knot with double points above. A different explanation is that the number of edges like the any given edge in , and hence the number of faces of configuration space whose variation is the contraction of that edge, is usually the number of automorphisms of .
The remaining faces of configuration space correspond to some number of vertices approaching each other. Most of these are shown to vanish in Chapter 5. The remaining faces are the anomalous faces: those faces in which the degenerating vertices are not connected to any other vertex by a propogator. (This includes the case in which all the vertices degenerate.) The variations resulting from these faces is described at length in Chapter 6, and is the source of the (potential) correction term in equation 2.4.
Chapter 3 Configuration space and its Compactification
Definition 3.1.
The configuration space of points indexed by the finite set in a smooth manifold is the subset of given by
(3.1) 
This is a smooth manifold, but it is not compact if the size of , , is greater than 1. The construction is functorial on the category of smooth manifolds and imbeddings, with, for an imbedding and ,
(3.2) 
which is in since is an imbedding.
I sometimes use the notation as a shorthand for when I need to refer to the points specifically by name.
The compactification of of appropriate for this context was first written down in an algebrogeometric context^{1}^{1}1Using the projective sphere bundle rather than the sphere bundle so the result is an honest manifold (or variety). by Fulton and Macpherson [10], and in the case of manifolds by Axelrod and Singer [2].
Section 3.1 aims to give an intuitive explanation and complete description of this wonderful compactification. Many details of the construction are deferred until Section 3.2, which is somewhat tangential to the main thrust of the thesis. It contains a new definition of the compactification. Section 3.3 describes some variations on the compactification necessary in our context.
3.1 A description of the compactification
Our basic requirement on the compactification is that the maps must extend smoothly to the compactification for all . Since the compactification should be compact, every path in must have a limit.^{2}^{2}2Actually, I won’t allow paths that approach spatial infinity for the moment, so “compactification” is a slight misnomer. See Section 3.3 for a description of the true compactification. Every such path has a welldefined limit in . Group into classes that map into the same point in in the limit. Then the direction between elements in different classes is determined by the limit of and ; so to guarantee that the corresponding extends continuously, the compactification will come with a map .
Now onsider a path in in which the points of just one subset of size greater than one approach the same point in . The with are completely determined by the relative configuration of the points in : the positions of the points in , modulo overall translation and scaling. This relative configuration comes with a natural map to the space of points whose coordinates are not all identical, modulo translation and scaling; this latter space is compact (it is the sphere , where is the diagonal in ), so every path has a limit point in it. So to help define the , those portions of in which the points of approach each other will come with a map to the space of relative configurations, which I’ll call .
But the relative configuration might not be enough: the relative configuration might approach a degenerate relative configuration, in which some set come together. This corresponds to all the points in approaching each other, but the points in approaching each other faster. But all the points of cannot come together in the limit, so we always gain something: will be strictly smaller than . So if the points in come together in , new coordinates, the relative configuration of the points in , will help define more of the .
In general, the coordinates of will depend on the particular point. The points in some subsets which approach each other at the top level (only counting maximal ), and the directions between points in different are determined by a macroscopic configuration, a point in . ( means with points in each identified.) Then, for each , the micrscopic configuration near the , the limit point in determine some more directions. Possibly some still approach each other; then the submicrscopic configuration at the , the limit point in determines yet more directions, and so forth.
The set of subsets that occur in this construction characterize the coordinates used. The valid are those which consist of subsets of of size for which every distinct pair satisfy one of
These characterize the faces of , which is a manifold with corners, a generalization of a manifold with boundary. (See below for a precise definition.) The reader should verify that the dimension of the face is , or codimension . ( is the number of distinct subsets of in .) The stratum correspond to is, intuitively, the set of limit points in the open configuration space in which points in each subset approach each other at comparable sppeds.
Although we won’t use the compactification for any spaces other than or , it’s worth pointing out that a similar compactification can be defined for any dimensional manifold . The one essential difference is that, without the Euclidean structure of , the relative configurations are defined only locally. Specifically, a coordinate chart near yields a homeomorphism from to a neighborhood of in the tangent space to at , , whose Jacobian at is the identity. In , the limiting configuration of the points in is well defined (taking values in : nonconstant maps from into the tangent space to at , modulo overall translation and scaling). The limit point in of a path in in which the points in all approach the point does not depend on the coordinate chart.
3.2 Precise definitions
It’s now time for some proper definitions. This section is somewhat off the main track of my thesis, and is somewhat independent of the other material. Statements are made in full generality that are not needed for these knot invariants. But the results are of some independent interest. In particular, I give a new and somewhat simpler definition of the compactification of configuration space.
As I mentioned, the compactification will be a manifold with corners.
Definition 3.2.
A (smooth) dimensional manifold with corners is a Hausdorff topological space covered by open sets, each homeomorphic to
(3.3) 
for some integer . The transition functions between two such open sets must extend to a smooth function on an open subset of .
This definition allows us to define smooth functions, smooth differential forms, the tangent space, etc. The general principle is that smooth functions (et al) are required to have a smooth extension to a neighborhood of the coordinate chart.
Any manifold with corners can be written as the disjoint union of dimensional manifolds, for , consisting of those points which have a neighborhood isomorphic to , with mapping to the origin. These spaces (or the connected components of them) are called the strata of .
The notion of blowup will be central in the construction of the space . Intuitively, the blowup of a manifold along a submanifold , remove from and glue in a replacement that records the directions points approach . The replacement is , meaning the sphere bundle of the normal bundle of . The normal bundle of is —that is, tangent vectors of paths approaching , modulo translation along itself. And the sphere bundle of a vector bundle is, over each point , . Thus removes paths that sit on and takes the quotient by the length of the tangent vector in .
Definition 3.3.
The blowup of a smooth manifold with corners along a closed imbedded submanifold with corners is the manifold with boundary that is with replaced by those points of that are actually the images of paths in . In particular, there is a natural smooth map and a partial inverse .
To construct a manifold diffeomorphic to this blowup explicitly, pick a Riemannian metric on and find an so that is a tubular neighborhood of . Then is diffeomorphic to our desired blowup.
If has a Riemannian structure, then can be identified with the unit vectors in that are perpendicular to .
Note the similarity with the coordinates on configuration space constructed above. What I called the “relative configuration” of points at is an element on the boundary of the blowup of along the main diagonal . The configuration space should therefore be obtained, in some sense, by blowing up along each of its diagonals , . But it’s impossible to do all the blowups at once, and successive blowups are slightly tricky. If you blow up along before , for a proper subset of , then is part of the boundary of the partial blowup. The blowup along can be done, but it’s easier to just require that be blown up before is.
With this restriction, when we blow up a diagonal it will always intersect the interior of the partial blowup, and we can define successive blowups using the proper transform: the proper transform of will be the closure of minus its subdiagonals within the partial blowup. Blowing up means blowing up the proper transform of .
Another worry is that the resulting configuration space might depend on the order chosen for the blowups, which would mean that it wouldn’t be truly functorial. For an example of blowups that don’t commute, see Figure 3.1 for the blowup of two intersecting lines in .
Another example that’s more relevant to our situation but harder to visualize is the blowup of three 2planes in intersecting at 0. (In fact, this is exactly what the intersection of the diagonals and in looks like, after taking the quotient by overall translation.) The blowup along the first two (in either order) will be isomorphic . In particular, the stratum lying over 0 is isomorphic to . But the third 2plane will intersect the boundary of the partial blowup only in the stratum lying over , and in some diagonal on the torus . Blowing up along this plane turns the stratum over 0 into a torus minus a diagonal, which isn’t very nice and won’t be symmetric.
With the restriction that all the for and be blown up before , all diagonals become disjoint before you get around to blowing them up. Blowups along disjoint submanifolds clearly commute (and can in fact be done simultaneously), so any permissible order of doing the blowups would yield the same result. But this is not true. For instance, after blowing up along each of the diagonals , , the diagonals and will still intersect. But this intersection seems somehow “nicer” than the nasty cases given above. This can be made precise.
Definition 3.4.
A collection , of submanifolds of a manifold intersect generically at a point if the map
(3.4) 
is surjective, where is the image of the diagonal imbedding of in and the map is the direct sum of the imbeddings of the in .
(The motivation is that this map is surjective exactly when the appropriate linear equations can be solved to find a common intersection point for small displacements of the ).
The submanifolds intersect generically if, for every in at least one of the , the such that intersect generically at .
Equivalently, the manifolds intersect generically at the point if and only if has a neighborhood such that
(3.5) 
for some manifolds and points , , so that . ( is the projection of onto the factor .)
Proposition 3.5.
If the , , are proper submanifolds of a manifold with corners that intersect generically, then successive blowups along the yield a result independent of the order.
Proof.
It’s enough to proceed locally, in the neighborhood of a point . For ease of notation, assume that all the intersect at . Choose a neighborhood of that can be written as in Definition 3.4 as , with . Then the blowup of along is , the blowup of this space along is , and so forth; after all the blowups, the result is , independently of the order they were preformed.
(The condition that the be proper submanifolds is only required because we can’t blow up a manifold along itself.) ∎
So the notion of generic intersection seems to correspond to intuition about “niceness” of the intersection. We’re ready to define the space , in stages from . At each stage, we’ve blown up some collection of diagonals of . The next stage is the blow up of any diagonal of for which all proper supersets of are contained in the . To prove the result doesn’t depend on the choice of which to blow up at each stage, it suffices to check that adjacent steps commute.
Proposition 3.6.
If the diagonals corresponding to all proper supersets of and have been blown up at some stage in the construction of , but not and , then and intersect generically.
Proof.
If , then their intersection at this stage will be empty. In this case, , which, by hypothesis, we’ve already blown up; and in the coordinates over , there exist points with either all the points in identical, or all the points in identical, but not both.
On the other hand, if , and will intersect, but the intersection will be generic.
at a given stratum, want to reduce to each subst. individually. Right: enough to show that each tangent factor is a product as we want. ∎
For any embedding , the diagonals of are the intersections of the diagonals of with , with a generic intersection. Since the sequence of blowups is the same in the two cases, the blowup will be functorial.
3.3 Further variations
The configuration spaces of interest to us differ from the spaces defined above in two ways. First, as previously mentioned, is not actually compact as is not compact. Instead of , the proper compactification is for some (fixed) point . , so , and is a proper compactification of . The maps extend continuously to this compactification as well; the direction a point approaches determines the direction from to points that do not approach , and the relative configuration at determines directions between points that approach .
Since points approaching turn out to be mostly irrelevant for the purposes of this thesis, in the future I’ll write to mean , leaving the strata at implicit.
The other difference is the presence of the knot. As explained above, this is a functorial construction, so a knot yields an imbedding (of manifolds with corners) of in , which defines for points on the knot and use these to construct certain integrals. But as explained in Section 1, for any but the simplest integrals this is not enough to construct an invariant. Most invariants involve integration of certain points over and other points over . To do this properly, we need the notion of a restricted configuration space of maps from to distinct points in , with the points in landing in the submanifold . (Note that points in range over the complement of the , , not the complement of the knot .) Its closure in , , is the desired compact space.
Since the knot is onedimensional, the spaces as I’ve defined them are not connected for . The (cyclic) order the points in occur around is significant. In this notation, will therefore be taken to include a cyclic order of its points.
It’s important to note that is smoothly fibered over the space of knots . is a subset of defined by equations like and and so is fibered over . Since is just the closure of in , it too is fibered over . The notation without qualifications will mean this bundle over .
Finally, since the codimension 1 strata of become important in Stokes’ theorem, a brief description is appropriate. As usual, the strata are parametrized by subsets . If consists of only free vertices, then the stratum can be written : that is, this stratum of configuration space is the product of the macroscopic configuration with the microscopic configuration.
On the other hand, if (so contains some points on the knot, necessarily a contiguous subset in the cyclic order on ), is fibered over , with fiber . This is again the macroscopic configuration and the microscopic configuration, but now the microscopic configuration is relative the line , the derivative of at the common limit of . This latter space is the intersection of the closure of with the stratum near the point (the common limit point of the points in ). Specifically, this is the space of nonconstant maps with points in in the subspace spanned by , modulo translations in the direction (so that points in remain in the right subspace) and overall scaling.
Chapter 4 Tying it Together
In this chapter I bring together a number of loose ends, completing the proof that the invariants I’ll construct will be Vassiliev (Section 4.1), explaining why the variation of an integral is the integral restricted to the boundary and how the same cancellations can prove that a number of different integrals are invariants and give the same value (Section 4.2), making some notes on the form of the integrals resulting (Section 4.3), completing the proof that the contributions from principal faces cancel for an integral corresponding to a weight diagram (Section 4.4), and introducing tinkertoy diagrams (Section 4.5).
4.1 Integrals are Vassiliev
We can define the configuration space of points on an immersed curve with double points: it is the closure in of , the space of distinct points on the immersed curve and points in space. This is a perfectly fine compact manifold with corners, so configuration space integrals make sense for curves with double points as well.
Given any configuration space integral on a curve with double points, suppose we approach a curve with double points by varying . As the strands near approach each other (with the tangents at and , and , remaining fixed), the submanifold of approaches a limiting position. The limiting position is not a smooth submanifold of . But there is an open dense subset of the limiting position that is smooth, and so the limit of the integral will be equal to the integral over this subset of the limiting position.
There are a number of connected components to this smooth portion of the limiting position, indexed by the set of points that get very close to the crossing, together with the subsets and that approach the crossing along the strands near and respectively. (Of necessity, and will each be connected subsets of and .) This portion of the limiting position then lies in the stratum of in which the points approach each other (at comparable speeds) and is the product of the macroscopic configuration space with a microscopic configuration space, whose details don’t matter much, but is the image of [configurations of points on a line in the direction , points on a line in the direction , and points in ] in .
See Figure 4.1 for a picture of the situation in the most relevant case, when has two points, one on each strand.
So the limiting integral can be broken into components with different subsets very near the crossing. But the integral over a component with 0 or 1 points near the crossing will clearly not depend on the direction of the crossing. Hence the st derivative of a knot invariant with crossings will be 0. This completes the proof of Proposition 2.2.
4.2 On the Pushforward
Our integrals are of the general form
(4.1) 
where is an form on , with . (Recall that is the fiber bundle over the space of knots whose fiber over is . For us, will be the pullback of a topdimensional form on some manifold , and so will be closed.) This type of expression is called integrating over the fiber or the pushforward.
Definition 4.1.
Given any form on a fiber bundle whose fibers are oriented dimensional manifolds with corners, the pushforward of along , denote , is the form on whose value on a chain is
(4.2) 
When is a smooth manifold (without boundary) the pushforward of a closed form will be closed. But when is a manifold with corners, this is not true; instead, we have
Proposition 4.2.
For bundle whose fibers are smooth oriented manifolds with corners with boundary (with the orientation induced from ),
(4.3) 
Proof.
For any chain in , we have
(4.4) 
using Stokes’ theorem and . ∎
This proposition justifies my claim that the variation of an integral is (up to sign) the integral restricted to the boundary: since the forms we integrate are closed^{1}^{1}1Even though the dimension of the fiber is equal to the degree of , this is not a trivial statement, since it requires to be closed as a form on the bundle. (being the pullback of a closed form), will be 0, and , the variation of the integral of , will be .
This proposition also tells us what happens as we vary the form . If we pick a form in the same cohomology class as , then for some form . Applying the above proposition, ; but is a form on and hence 0. Thus the difference between the integrals of and will be for some form .
Therefore an integral will both

be an invariant under deformation of the knot and

be independent of the form within a cohomology class
if the boundary of the bundle is 0 in some sense. More precisely, since the forms we use are all for some form on a manifold (which will be a product of 2spheres) the image of the boundary of in this product should be 0 (as an chain). But since there is not yet any sensible homology theory with general manifolds with corners, I’ll state this in a different form.
Precisely, the arguments I’ll use to show the vanishing or cancellation of the pushforward of along a given nonanomalous face