Configuration Space Integrals Research Articles

The Fox–Hatcher cycle and a Vassiliev invariant of order three

We show that the integration of a 1-cocycle I(X) of the space of long knots in R^3 over the Fox-Hatcher 1-cycles gives rise to a Vassiliev invariant of order exactly three. This result can be seen as a continuation of the previous work of the second named author, proving that the integration of I(X) over the Gramain 1-cycles is the Casson invariant, the unique nontrivial Vassiliev invariant of order two (up to scalar multiplications). The result in the present paper is also analogous to part of Mortier's result. Our result differs from, but is motivated by, Mortier's one in that the 1-cocycle I(X) is given by the configuration space integrals associated with graphs while Mortier's cocycle is obtained in a combinatorial way.

Generalized Bott–Cattaneo–Rossi invariants of high-dimensional long knots

Bott, Cattaneo and Rossi defined invariants of long knots $\mathbb{R}^{n} \hookrightarrow \mathbb{R}^{n+2}$ as combinations of configuration space integrals for $n$ odd $\geq 3$. Here, we give a more flexible definition of these invariants. Our definition allows us to interpret these invariants as counts of diagrams. It extends to long knots inside more general $(n+2)$-manifolds, called asymptotic homology $\mathbb{R}^{n+2}$, and provides invariants of these knots.

Diagram Complexes, Formality, and Configuration Space Integrals for Spaces of Braids

AbstractWe use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen’s iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.

Milnor invariants of string links, trivalent trees, and configuration space integrals

We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent “homotopy link diagrams” which corresponds to all finite type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas for products of Milnor invariants from trivalent forests. We also obtain cohomology classes in spaces of link maps from the same data.

A formula for the Θ–invariant from Heegaard diagrams

The Theta invariant is the simplest 3-manifold invariant defined with configuration space integrals. It is actually an invariant of rational homology spheres equipped with a combing over the complement of a point. It can be computed as the algebraic intersection of three propagators associated to a given combing X in the 2-point configuration space of a Q-sphere M. These propagators represent the linking form of M so that $\Theta(M,X)$ can be thought of as the cube of the linking form of M with respect to the combing X. The Theta invariant is the sum of $6 \lambda(M)$ and $p\_1(X)/4$, where $\lambda$ denotes the Casson-Walker invariant, and $p\_1$ is an invariant of combings that is an extension of a first relative Pontrjagin class. In this article, we present explicit propagators associated with Heegaard diagrams of a manifold, and we use these "Morse propagators," constructed with Greg Kuperberg, to prove a combinatorial formula for the Theta invariant in terms of Heegaard diagrams.

On volume-preserving vector fields and finite-type invariants of knots

We consider the general non-vanishing, divergence-free vector fields defined on a domain in$3$-space and tangent to its boundary. Based on the theory of finite-type invariants, we define a family of invariants for such fields, in the style of Arnold’s asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.

The Oriented Graph Complexes

Oriented graph complexes, in which graphs are not allowed to have oriented cycles, govern for example the quantization of Lie bialgebras and infinite dimensional deformation quantization. It is shown that the oriented graph complex GC^or_n is quasi-isomorphic to the ordinary commutative graph complex GC_{n-1}, up to some known classes. This yields in particular a combinatorial description of the action of the Grothendieck-Teichm\"uller Lie algebra on Lie bialgebras, and shows that a cycle-free formality morphism in the sense of Shoikhet can be constructed rationally without reference to configuration space integrals.

CONFIGURATION SPACE INTEGRALS AND THE COHOMOLOGY OF THE SPACE OF HOMOTOPY STRING LINKS

Configuration space integrals have been used in recent years for studying the cohomology of spaces of (string) knots and links in ℝn for n > 3 since they provide a map from a certain differential graded algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links — the space of smooth maps of some number of copies of ℝ in ℝn with fixed behavior outside a compact set and such that the images of the copies of ℝ are disjoint — even for n = 3. We further study the case n = 3 in degree zero and show that our integrals represent a universal finite type invariant of the space of classical homotopy string links. As a consequence, we deduce that Milnor invariants of string links can be written in terms of configuration space integrals.

1-loop graphs and configuration space integral for embedding spaces

AbstractWe will construct differential forms on the embedding spaces Emb(j, n) for n − j ≥ 2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(j, n) by Emb(j, n), the homotopy fiber of the inclusion Emb(j, n) ↪ Imm(j, n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(j, n) and Emb(j, n). In particular we obtain nontrivial cohomology classes (for example, in H3(Emb(2, 5))) of higher degrees than those of the first nonvanishing homotopy groups.

COMBINATORIC MASSEY–MILNOR LINKING THEORY

In this paper following the scheme of Massey–Milnor invariant theory [C. C. Hsieh, Combinatoric and diagrammatic studies in knot theory J. Knot Theory Ramifications16 (2007) 1235–1253; C. C. Hsieh, Massey-Milnor linking = Chern-Simons-Witten graphs, J. Knot Theory Ramifications17 (2008) 877–903; C. C. Hsieh and S. W. Yang, Chern-Simons-Witten configuration space integrals in knot theory, J. Knot Theory Ramifications14 (2005) 689–711], we studied the first non-vanishing linkings of knot theory in ℝ3 and also derived the combinatorial formulae from which we could read out the invariants directly from the knot diagrams. Though the theme is calculus, the idea comes from perturbative quantum field theory.

An integral expression of the first nontrivial one-cocycle of the space of long knots in ℝ3

Our main object of study is a certain degree-one cohomology class of the space K of long knots in R^3. We describe this class in terms of graphs and configuration space integrals, showing the vanishing of some anomalous obstructions. To show that this class is not zero, we integrate it over a cycle studied by Gramain. As a corollary, we establish a relation between this class and (R-valued) Casson's knot invariant. These are R-versions of the results which were previously proved by Teiblyum, Turchin and Vassiliev over Z/2 in a different way from ours.

Borromean rings and linkings

For links of 3 components, such as Borromean rings, which escape the detection of Gauss linking, we define and compute combinatorically and explicitly the higher linking. And as in perturbative quantum field theory, this higher linking is presented as a sum of Chern–Simons–Witten configuration space integrals.

On Kontsevich's characteristic classes for higher-dimensional sphere bundles II: Higher classes

This paper studies Kontsevich’s characteristic classes of smooth bundles with fibre in a ‘singularly framed’ odd-dimensional homology sphere, which are defined through his graph complex and configuration space integral. We will give a systematic construction of smooth bundles parameterized by trivalent graphs and will show that our smooth bundles are non-trivially detected by Kontsevich’s characteristic classes. It turns out that there are surprisingly many non-trivial elements of the rational homotopy groups of the diffeomorphism groups of spheres that are in some ‘non-stable’ range. In particular, the homotopy groups of the diffeomorphism groups in some ‘non-stable’ range are not finite.

MASSEY–MILNOR LINKING = CHERN–SIMONS–WITTEN GRAPHS

We express the first non-vanishing Massey–Milnor linkings in terms of Chern–Simons–Witten configuration space integrals in perturbative quantum field theory.

Configuration space integral for longn–knots and the Alexander polynomial

There is a higher dimensional analogue of the perturbative Chern–Simons theory in the sense that a similar perturbative series as in 3 dimensions, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott– Cattaneo–Rossi invariant). This invariant was constructed by Bott for degree 2 and by Cattaneo–Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n–knots and characterize the Bott–Cattaneo–Rossi invariant as a finite type invariant of long ribbon n–knots introduced by Habiro– Kanenobu–Shima [10]. As a consequence, we obtain a nontrivial description of the Bott–Cattaneo–Rossi invariant in terms of the Alexander polynomial.

A SURVEY OF BOTT–TAUBES INTEGRATION

It is well-known that certain combinations of configuration space integrals defined by Bott and Taubes [11] produce cohomology classes of spaces of knots. The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail. The aim of this paper is to fill in the gaps as well as summarize the importance of these integrals.

Configuration space integrals and Taylor towers for spaces of knots

We describe Taylor towers for spaces of knots arising from Goodwillie–Weiss calculus of the embedding functor and extend the configuration space integrals of Bott and Taubes from spaces of knots to the stages of the towers. We show that certain combinations of integrals, indexed by trivalent diagrams, yield cohomology classes of the stages of the tower, just as they do for ordinary knots.

CHERN–SIMONS–WITTEN CONFIGURATION SPACE INTEGRALS IN KNOT THEORY

In the context of perturbative quantum field theory, we express the first non-vanishing Massey–Milnor linkings in terms of Chern–Simons–Witten configuration space integrals in knot theory.

The configuration space integral for links in ℝ3

The perturbative expression of Chern-Simons theory for links in Euclidean 3-space is a linear combination of integrals on configuration spaces. This has successively been studied by Guadagnini, Martellini and Mintchev, Bar-Natan, Kontsevich, Bott and Taubes, D. Thurston, Altschuler and Freidel, Yang and others. We give a self-contained version of this study with a new choice of compactification, and we formulate a rationality result.

Configuration spaces and Vassiliev classes in any dimension

The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.