The Homology Theory of Graphs and The Configuration Space Integral

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How to classify all knots is an important problem that has been puzzling mathematicians since the end of the 19th century. Many of them have mostly attempted to use knot invariant theory to prove the possibility of their classification. This shows that the construction and research of knot invariant is a crucial point to the solution to this confusing problem. A knot invariant is a knot function which is invariant under the isotopy deformation. The ``linking number” is a very typical example. Later, the Chern-Simons theory which is proposed by Chern and Simons in 1971 is the most popular example of topological field theory in 3 dimensions. Given a compact Lie group G, a compact, oriented 3-manifold M, a link L in M, and a representation of G for each component of L, this theory associates some topological invariants with these data. There are several ways to define the invariants, which are all closely related to one another. First of all there are the non-perturbative methods by Witten, Reshetikhin and Turaev: Witten [W] used fundamental properties of quantum field theory, in particular the path integral formulation, and Reshetikhin, Turaev [RT] used quantum groups. These two definitions are equivalent to each other. Around 1989, there are many people working on the perturbative approach. The first of them are given by Guadagnini, Martellini and Mintchev [GMM] in the case M=S3, L nonempty, using propagators and the trivalent graphs which are composed of base points on the L, inner points in R3 and all of the inner points are trivalent. This approach was then elaborated by Bar-Natan [B1], [B2]. The case M being not S3, L = ф was treated by Axelrod and Singer [AS]. A common feature of all the work is the trivalent graphs expansion familiar in perturbative quantum field theory. Invariants are defined at very order in the expansion, where the order of a graph is defined to be the number of edges minus the number of inner points of the graph, each is a formal sum of several terms corresponding to the graphs of the given order. The contribution of any graph is the product of two factors, the first depends only on the group G and the representations associated to the components of L, and the second is independent of G and its representations, it is an integral over the configuration space of the vertices of the graph, some of which are constrained to lie on L, which the others can lie anywhere in the complement of L. When L is a knot in R3, several properties of the invariant Arising from the contributions of order 2 were discussed in [GMM], and there is example that Guadgnini, Martellini, and Mintchev derived a formula about the ``self-linking number" of knots according to the Perturbative Chern-Simons theory (for order 2). Besides, Bar-Natan also proved it in his article presented in 1995. During the same period, Vassiliev studied the topology of the complement of knot space in smooth function space and used the Vassiliev Complex to express its topology construction. It was showed that the homology group of the Vassiliev complex can produce knot invariants. The subject of Vassiliev knot invariant, also known as finite type invariant, was developing rapidly. The starting point of Vassiliev [V] was the space of all immersions of S1 in S3. In this space, a knot type is a cell whose faces are singular knots with a finite number of transversal double points. Any knot invariant can be extend to such singular knots. It is said to be a finite type invariant of order smaller N, if it vanishes on all singular knots with more than N double points. Let VN be the space of invariants of order smaller N, unexpectedly at first, Bar-Natan found that VN/VN-1 embeds in the dual of the space of BN diagrams of degree N, where the BN diagrams are proposed by Bar-Natan in [B1], [B2]. And Kontsevich showed that these two spaces are in fact isomorphic. Bar-Natan involved the construction of a universal Vassiliev invariant, a formal power series in the space of BN diagrams whose coefficients are finite type invariants, based on the Knizhnik-Zamolodchikov equations of the WZW modle of conformal field theory. Later, in 1991, Kontsevich and Bar-Natan (independently) combined the Perturbative Chern-Simons theory with Vassiliev knot invariant and accomplished configuration space integrals of the R3 space graphs of any order; that s, Z(K). Then, in 1993, Kontsevich adopted the iterated integral theory of Chen Kuo-Tsai, and Cauchy form in the integral, he finally obtained the Kontsevich integral Z(K), which was soon proved to be the `` Universal Vassiliev invariant", also known as universal Quantum Invariant. And so, the framework on Quantum Invariant initiated by Jones since 1984 had been combined. However, in 1994, there was a breakthrough on the theory of configuration space.Bott and Taubes [BT] used the construction of compactification of the configuration space due to W. Fulton and Mac Pherson [FP], to get a compacted manifold with corners and to prove finiteness of the configuration space integral. In order to show that the contributions of graphs of a given order summed up to an invariant, we must compute the variations of these integrals under a small change of the embedding of L, and this was proved to be quite difficult and lengthy. However, Bott and Taubes [BT] indeed succeeded in doing this. They showed that the variations could be split into two terms, the ``diagrammatic" and ``anomalous" variations. As its name indicates, the diagrammatic variation can be read at once from the trivalent graphs. It corresponds to the differential of the Kontsevich graph complex, obtained by collapsing the edges. The anomalous variation is more difficult to compute, but it is proportional to the variation of the first order contribution, the ``self-linking number". The constant of the proportionality, is still unknown in general, but independent of the embedding. And then Bott and Taubes constructed a universal Vassiliev knot invariant, given by the perturbative expansion of the expectation value of a Wilson loop in Chern-Simons theory on R3. The basic ingredient in the integrals obtained from the Feynman rules is the propagator of the gauge by the Gauss two-form, the pullback of the volume form on S2. After thorough analyses, Bott and Taubes also found there still existed some anomaly boundaries that couldn't be neglected beside the principal boundary considered by Kontsevich and Bar-Natan. Therefore, they presented the anomalous boundaries, which were derived when graphs concentrated on a certain point of the knot. These graphs are called infinitesimal graphs on knots. Bott and Taubes conjectured that the associated integral of the anomalous boundaries could be non-zero. But, Prof. Yang Su-Win used the degree theory to prove that the anomaly of the graphs of order 3 is zero. This way, the zero anomaly of order 3 is proved. Accordingly, we learned the presumption of Bott and Taubes was correct to a certain degree, but the result, zero of Yang, derived from the combination of the two parts, was far beyond their consideration. It was not until 1995 that Altschuler and Freidel used the marvelous findings of Bott-Taubes and Bar-Natan and made a further study on it. And then they found that by slightly adjusting the configuration space integral Z(K), (or we can say by multiplying an adjusted item), they obtained a framed knot invariant. The adjusted item was called the anomaly; that is, they reached a conclusion for a framed knot invariant. In addition, Altschuler and Freidel also proved the following results: (a). If a graph is even order, then the integral of the anomalous boundary of the graph is zero. (b). If a graph is not primitive, then the integral of the anomalous boundary of the graph is zero, where ``primitive" means that a graph is connected when we take away the support of the graph. These results, when put together with Prof. Yang's computation, led to the fact: The anomaly of all graphs of order 3 is zero. So far, this is all we know about anomaly. By the year 1999, S. Poirier made use of Prof. Yang's degree theory, and tried to extend his study of the framed knot invariants of Altschuler and Freidel to link invariants. He put forth his theory of the limit configuration space integral and proposed some results. So far, there have been still two major open problems concerning the configuration space integrals and the Perturbative Chern-Simon theory, Open problem A: Is Z(K) sufficient to classify all the knots? Open problem B: Whether or not the universal Vassiliev invariants derived from the limit configuration space integral and the Kontsevich integral are equal? As we know, open problem A has been a tough problem that remains to be solved, so we would just skip it this moment; instead, we will concentrate on the open problem B. However, we have learned that the most obvious difference between the configuration space integral and the Kontsevich integral lies in the fact that one uses the Gaussian form ( 2-form ), but the other uses the Cauchy form ( 1-form). On the other hand, one uses all the chords of the trivalent graphs in the R3 space, but the other one considers only the horizontal chords. Due to the result of S. Poirier, if we can prove the zero anomaly, we are sure to obtain the result Zl(X)=exp(H/2), in which H refers to one chord graph with the support of two parallel lines and ``X” is the graph of 2-braids. And then by combining this with the result of T. Q. T. Le and J. Murakami [LM], we obtain the equality in open problem B. Nevertheless, the zero anomaly has long been a puzzle to Bott and Taubes, and it seems to be difficult to prove it in a direct way. Therefore, The motivation and main purpose of this dissertation is to clear the doubt of the zero anomaly in Bott and Taubes' mind. This dissertation will mainly focus on the discussion of open problem B. And we know the solution to open problem B lies in clarification of the zero anomaly. To deal with the zero anomaly, we may well propose the graph homology theory, in which all graphs are on the support of two parallel lines. By doing this, we hope to prove the acyclic property of the chain complex of the special graphs and the special graphs are the graphs which have two parallel lines as support. This is the reason for us to study the graphs with two parallel lines as support, which are the so-called B2-graphs as above. We also prove the Cycle Theorem, and the Cycle Theorem will imply the result Zl (X)=exp(H/2). And this result corresponds to that of Poirier [P] and [LM]; meanwhile, we also have the zero anomaly property proved and Open Problem B clarified. This is, in other words, to kill two birds with one arrow. Thus, the acyclic property of the R3 graphs of all orders is the key thing in our work. In order to deal with the R3 space graphs, we consider the projection of the R3 space graphs onto the plane and we get the R2 plane graphs. In this way, the configuration space of the R3 space graph has a structure of fiber bundle, and the base space is the corresponding configruation space of plane graph, and the fibre H(Г) is the linear space of all the height functions. By tranforming the R3 space graphs to the R2 plane graphs, we could prove the Transfer Principle of the quotient chain complex in the R3 space graphs. So that we also prove the equivalence of the acylic property of both the R3 space graphs and the R2 plane graphs. Furthermore, we extend our study from space graphs to block graphs, in which `` block" is formed by the open subsets that appear when the height function space H(Г) is cut by some hyperplanes determined by the simple cycles. We will make use of these block graphs to construct a chain complex and to prove the zero anomaly of the graphs of order 3. To reach the aim of our study, we divide the edges of the plane graphs into two classes; one is the class of outer edges, each of which joins a base point and an inner point, and the other consists of inner edges, each of which joins two distinct inner points. In doing this we should first define the edge-contraction boundary operator ∂E on the plane graphs. And next we divide the boundary operator ∂E into ∂I and ∂O, which individually refer to inner-edge contraction boundary operator and outer-edge contraction boundary operator. We could define the chain complex of the plane graphs under the boundary operator ∂I and the chain complex is denoted by C2={Cpn, ∂I}, where the Cpn is the real vector space generated by all oriented plane graphs with p edges of order n. We also clearly define the degree on the plane graphs. In this way, we obtain the filtration according to the degree we defined. With this filtration, we get a new spectral sequence. However, in order to make comparison with the construction of differential forms on graphs of Bott and Taubes, that is, the key property dω(Г)=ω(∂ (Г)), we have to prove a result that only the top dimension homology group remains when taking ∂I -homology. This is the trivalent conjecture we proposed. With that, we prove H*(C2)=H*(K2), in which C2 refers to the original chain complex as stated above. For the chain complex K2, we give the following explanation: By the trivalent conjecture, we obtain the top dimension homology groups, say Kpn generated by the cycles under the boundary operator ∂I which are constructed by the plane graphs with p edges of order n. And we define the chain complex K2 to be { Kpn, ∂O}. Of course, we have identified the truth of the trivalent conjecture of order 5 by using the graph homology theory; meanwhile, we make sure the plane graphs of order 5 are qualified with acyclic property. This is one of the main results of this dissertation. We continuously avail the acyclic property of plane graphs in order 5 to deal with the space graphs that we really desire. In doing this, we have to divide the boundary operator into three parts: ∂I, ∂O, and a new addition ∂B. Among these, ∂B represents the boundary operator action on the open parts of the space H(Г). Therefore, we obtain a chain complex with three different directions. The main purpose of this is to find those trivalent R3 graphs whose boundaries are the given cycle. Though the existence of those trivalent R3 graphs is still undoubted, the trivalent R3 graphs may not be trivalent. That's why we devoted so much time and thought to searching for the trivalent R3 graphs for the sake of the corresponding to the rule of Bott and Taubes, dω(Г)=ω(∂ (Г)). Thus, we have the existence of trivalent R3 graphs proved, and we also complete the work on the trivalent acyclic property of order 5 on the R3 space graphs. This is a crucial step to prove the zero anomaly of order 5. Moreover, we apply the theory of block graphs to classify all block graphs on the anomaly plane graphs of order 3, and we divide ∂’ into two parts; that is, ∂’=∂B+ ∂E, in which ∂E=∂I+∂O, then we derive a double chain complex and prove the zero anomaly of order 3. In conclusion, we want to state that we apply a procedure completely different from the degree theory to prove that order 3 is a zero anomaly. This also shows graph homology theory is more preferable in application. In next part, we would generally divide my dissertation into eight sections as follows: In section 1, we propose a new ``edge-direction Orientation “to define the oriented graphs and rewrite the AS, IHX, STU, IC four relations for the real vector space A(M) which is generated by oriented graphs. We also adopt the results of the configuration space ntegral completed by some mathematical workers, and add to them some supplements to enhance its coherence. Meanwhile, we use that as the introductory statement of this essay to describe the main purpose of this study. And in section 2, we introduce the Bn-graphs of R3 and the plane graphs of R2, added with boundary operator of the plane graphs. Furthermore, the hyperplane system and the height function space are also introduced to serve as important foundations to prove the zero anomaly of order 3 in section 8. In section 3, we prove the Transfer Theorem E of the quotient chain complex and that is what we use to prove the zero anomaly of order 3, and according to that, we deliberately ignore the plane graphs of non-orientable instead of discussing it when we classify the plane graphs. Next, in section 4, we not only detail the close relation between B2-graphs and the anomaly, but introduce the double chain complex of the plane graphs. With these, we obtain the spectral sequence of the filtered complex C2. At the same time, we take the trivalent conjecture and prove that H*(K2)=H*(C2). The main part of section 5 is focused on the calculation of the homology groups of the plane graphs of order 5. First, we apply Vassiliev diagram to classify all the plane graphs of order 5. Next, we divide the boundary operator into the inner-edge and the outer-edge boundary operators. By doing that, we obtain a leveled chain complexes under the inner-edge boundary operator. We continuously compute the row sequences and find out that there only homology groups in the top dimension survive. And then we go on computing the homology groups under the outer-edge boundary operator in the first column from the left side. With all the work done, the acyclic property of the plane graphs of order 5 has been proved. However, this study is chiefly aimed at discussing problems of the R3 space graphs; therefore, in section 6, we apply the chain complex of three boundary operators to further prove the existence of the trivalent graphs by combining the Global Acyclic Theorem we state. Thus comes the result dω(Г)=ω(∂ (Г)), which, supported by the acyclic property of the plane graphs of order 5, in turn derives the acyclic property of the R3 space graphs of order 5 and this completes the zero anomaly of order 5. The main job of section 7 is to once again justify the property of the leftover in the top dimension. By using the way mentioned above in section 5, we only consider the plane graphs, with 2 outer-edges joining to the left base point and 3 outer-edges joining to the right base point, which are called (2,3)-graphs. We also define the following two classes of the plane graphs which are called inner-stretch graphs and inner-splitting graphs. In order to relax the computation of the plane graphs, we prove the above two classes of the plane graphs can be omitted. Then by figuring the plane (2,3)-graphs of order 6, we compute the sequence of the (2,3)-graph of order 6, and we once again prove it with certitude. In the final part, section 8, we use the foundations, described in section 2 and 3, to verify the zero anomaly of order 3. First, we explicate the action of Aut(Г) on the open block parts of H(Г). Then we further classify all the block graphs, and divide the boundary ∂’ into ∂B and ∂E, and calculate the double chain complex of the block graphs. After these, we make sure that only the homology group of the top dimension is left over on the top left side. However, the block graphs of the anomaly of order 3 are situated in the space of the next stage which neighbors with the top left side. According to this, we prove the zero anomaly of order 3, because, except the homology, no other thing is seen left on the top left side. This is the important method, completely different from the degree theory, that we use to prove the zero anomaly of order 3.