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.
Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct "Vassiliev classes" in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in Euclidean spaces of dimension greater than three. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configurati…
The $C_k$-equivalence is an equivalence relation generated by $C_k$-moves defined by Habiro. Habiro showed that the set of $C_k$-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order $\leq k-1$. We see that the set of $C_k$-equivalence classes of the spatial $θ$-curves forms a group under the vertex connected sum and that if the group is abelian, then it can be classified by the additive Vassiliev invariant of order $\leq k-1$. However the group is not necessarily abelian. In fact, we show that it i…
This work continues the study of a homotopy-theoretic construction of the author inspired by the Bott-Taubes integrals. Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots. Their techniques were later used by Cattaneo et al. to construct real "Vassiliev-type" cohomology classes in spaces of knots in higher-dimensional Euclidean space. By doing this integration via a Pontrjagin-Thom construction, we constructed cohomology classes in the knot space with arbitrary coefficients. We later showed that a refinement of this …
We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for this invariant.
We give an introductory survey on the universal Vassiliev invariant called the perturbative series expansion of the Chern-Simons theory of links in euclidean space, and on its relation with the Kontsevich integral. We also prove an original geometric property of the anomaly of Bott, Taubes, Altschuler, Freidel and D. Thurston, that allowed Poirier to prove that the Chern-Simons series and the Kontsevich integral coincide up to degree 6.
We find that Koschorke's $β$-invariant and the triple $μ$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin suspension, and both were unexpected for the author, because the only known direct ways to extract $μ$ and $β$ from invariants of maps between configuration spaces involved some homotopy theory (Whitehead products and the stable Hopf invariant, respectively).
Given a flexible $n$-gon with generic side lengths, the moduli space of its configurations in $\mathbb{R}^2$ as well as in $\mathbb{R}^3$ is a smooth manifold. It is equipped with $n$ \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over $\mathcal{M}_{0,n}$. We study their Euler classes, first Chern classes and intersection numbers, that is, top monomials in Chern (Euler) classes. The latter are interpreted geometrically as the signed numbers of some \textit{triangular configurations} of the flexible polygon.
We are interested in comparing properties of symplectic mapping class groups of symplectic manifolds of dimension four or higher with properties of classical mapping class groups of surfaces. For $n \geq 2$, consider a configuration of Lagrangian $S^n$s in a Weinstein domain $M^{2n}$. If it is analogous, in some sense that we make precise, to a configuration of exact Lagrangian $S^1$s on a surface $Σ$, we show that any relation between Dehn twists in the $S^n$s must also hold between the $S^1$s. Such analogous pairs of configurations include plumbings of $T^\ast S^1$s and $T^\ast S^n$s with th…
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.
We will construct differential forms on the embedding spaces Emb(R^j,R^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(R^j,R^n) by fEmb(R^j,R^n), the homotopy fiber of the inclusion Emb(R^j,R^n) -> Imm(R^j,R^n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(R^j,R^n) and fEmb(R^j,R^n). In particular we obtain nontrivial c…
Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by Lagrangian submanifolds. The main result proves that the $N^{th}$ coloured Jones polynomial and $N^{th}$ coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in ea…
We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration.
In this paper we determine the topological complexity of configuration spaces of graphs which are not necessarily trees, which is a crucial assumption in previous results. We do this for two very different classes of graphs: fully articulated graphs and banana graphs. We also complete the computation in the case of trees to include configuration spaces with any number of points, extending a proof of Farber. At the end we show that an unordered configuration space on a graph does not always have the same topological complexity as the corresponding ordered configuration space (not even when they…
For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$ such that images of every pairwise distinct $k$ points are affinely (linearly) independent. Such maps are called affinely (linearly) $k$-regular embeddings. We investigate the cohomology obstructions to existence of regular embeddings and give some new lower bounds on the dimension $m$ as function of $X$ and $k$, for the cases $X$ is $\mathbb R^n$ or $X$ is an $n$-dimensional manifold. In the latter case, some nonzero Stiefel--Whitney classes of $X$ help to improve the bound.
We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this "integration" homotopy-theoretically, we are able to produce integral cohomology classes. We then show how this integration is compatible with the homology operations on the space of long knots, as studied by B…
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is 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 in [HKS]. As a consequence, we obtain a …
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 fo…
Let $X$ be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle $π$ is the identity. This space $X$ is closely related to the PSL$_2(\mathbb{R})$-character variety of the genus 2 surface $Σ$. In this article we study the topology and the natural symplectic structure on $X$, and we describe the action of the mapping class group of $Σ$ on $X$. This completes the classification of the ergodic components of the character variety in genus 2 initiated in our previous work.
In this paper we construct a topological model for the Witten-Reshetikhin-Turaev invariants for $3$-manifolds coming from the quantum group $U_q(sl(2))$, as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level $\cN \in \N$ we show that the level $\cN$ WRT invariant for a $3-$manifold is a state sum of Lagrangian intersections in a covering of a {\bf fixed} configuration space in the punctured disk. This model brings a new perspective on the structure of the level $\cN$ Witten-Reshetikhin-Turaev invariant, showing that it is completely enco…
