Searcharxiv⌕ Search

arXiv subjects

Christine Lescop

Publications and source records attributed to Christine Lescop.

16 recordsLinked to original sources
arXiv2024

Invariants of links and 3-manifolds from graph configurations

In this self-contained book, following Edward Witten, Maxim Kontsevich, Greg Kuperberg and Dylan Thurston, we define an invariant Z of framed links in rational homology 3-spheres, and we study its properties. The invariant Z, which is often called the perturbative expansion of the Chern-Simons theory, is valued in a graded space generated by Jacobi diagrams. It counts embeddings of this kind of unitrivalent graphs in the ambient manifold, in a sense that is explained in the book, using integrals over configuration spaces, or, in a dual way, algebraic intersections in the same configuration spaces. When the ambient manifold is the standard 3-sphere, the invariant Z is a universal Vassiliev link invariant studied by many authors including Guadagnini, Martellini and Mintchev, Bar-Natan, Bott and Taubes, Altschüler and Freidel, Thurston and Poirier... This book contains a more flexible definition of this invariant. We extend Z to a functor on a category of framed tangles in rational homology cylinders and we describe the behaviour of this functor under various operations including some cabling operations. We also compute iterated derivatives of our extended invariant with respect to the discrete derivatives associated to the main theories of finite type invariants. Together with recent results of Massuyeau and Moussard, our computations imply that the restriction of Z to rational homology 3-spheres (equipped with empty links) contains the same information as the Le-Murakami-Ohtsuki LMO invariant for these manifolds. They also imply that the degree one part of Z is the Casson-Walker invariant.

math.GT↗
arXiv2024

On elementary invariants of genus one knots and Seifert surfaces

This elementary article introduces easy-to-manage invariants of genus one knots in homology 3-spheres. To prove their invariance, we investigate properties of an invariant of 3-dimensional genus two homology handlebodies called the Alexander form. The Alexander form of a 3-manifold E with boundary contains all Reidemeister torsions of link exteriors obtained by attaching two-handles along the boundary of E. It is a useful tool for studying Alexander polynomials and Reidemeister torsions. We extract invariants of genus one Seifert surfaces from the Alexander form of their exteriors.

math.GT↗
arXiv2015

A formula for the Theta 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 $Θ(M,X)$ 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 $6 λ(M)$ and $p\_1(X)/4$, where $λ$ 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.

math.GT↗
arXiv2015

An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations

These introductory lectures show how to define finite type invariants of links and 3-manifolds by counting graph configurations in 3-manifolds, following ideas of Witten and Kontsevich. The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. As an important example, we present it as the algebraic intersection of a torus and a 4-chain called a propagator in a configuration space. In the second section, we introduce the simplest finite type 3-manifold invariant, which is the Casson invariant (or the Theta-invariant) of integer homology 3-spheres. It is defined as the algebraic intersection of three propagators in the same two-point configuration space. In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams. In Sections 4 and 5, we sketch an original construction based on configuration space integrals of universal finite type invariants for links in rational homology 3-spheres and we state open problems. Our construction generalizes the known constructions for links in the ambient space, and it makes them more flexible. In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.

math.GT↗
arXiv2014

A combinatorial definition of the Theta-invariant from Heegaard diagrams

The invariant $Θ$ is an invariant of rational homology 3-spheres $M$ equipped with a combing $X$ over the complement of a point. It is related to the Casson-Walker invariant $λ$ by the formula $Θ(M,X)=6λ(M)+p_1(X)/4$, where $p_1$ is an invariant of combings that is simply related to a Gompf invariant. In [arXiv:1209.3219], we proved a combinatorial formula for the $Θ$-invariant in terms of Heegaard diagrams, equipped with decorations that define combings, from the definition of $Θ$ as an algebraic intersection in a configuration space. In this article, we prove that this formula defines an invariant of pairs $(M,X)$ without referring to configuration spaces, and we prove that this invariant is the sum of $6 λ(M)$ and $p_1(X)/4$ for integral homology spheres, by proving surgery formulae both for the combinatorial invariant and for $p_1$.

math.GT↗
arXiv2013

A universal equivariant finite type knot invariant defined from configuration space integrals

In a previous article, we constructed an invariant Z for null-homologous knots in rational homology spheres, from equivariant intersections in configuration spaces. Here we present an equivalent definition of Z in terms of configuration space integrals, we prove that Z is multiplicative under connected sum, and we prove null Lagrangian-preserving surgery formulae for Z. Our formulae generalize similar formulae that are satisfied by the Kricker rational lift of the Kontsevich integral for null Borromean surgeries. They imply that Z is universal with respect to a natural filtration. According to results of Garoufalidis and Rozansky, they therefore imply that Z is equivalent to the Kricker lift of the Kontsevich integral for null-homologous knots with trivial Alexander polynomial in integral homology spheres.

math.GT↗
arXiv2013

On homotopy invariants of combings of 3-manifolds

Combings of oriented compact 3-manifolds are homotopy classes of nowhere zero vector fields in these manifolds. A first known invariant of a combing is its Euler class, that is the Euler class of the normal bundle to a combing representative in the tangent bundle of the 3-manifold $M$. It only depends on the Spin$^c$-structure represented by the combing. When this Euler class is a torsion element of $H^2(M;Z)$, we say that the combing is a torsion combing. Gompf introduced a $Q$-valued invariant $θ_G$ of torsion combings of closed 3-manifolds that distinguishes all combings that represent a given Spin$^c$-structure. This invariant provides a grading of the Heegaard Floer homology $\hat{HF}$ for manifolds equipped with torsion Spin$^c$-structures. We give an alternative definition of the Gompf invariant and we express its variation as a linking number. We also define a similar invariant $p_1$ for combings of manifolds bounded by $S^2$. We show that the $Θ$-invariant, that is the simplest configuration space integral invariant of rational homology spheres, is naturally an invariant of combings of rational homology balls, that reads $(\frac14p_1 + 6 λ)$ where $λ$ is the Casson-Walker invariant. The article also includes a mostly self-contained presentation of combings.

math.GT↗
arXiv2010

Knot invariants derived from the equivariant linking pairing

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M;Z)/Torsion. We present the invariant Q(M,K) and some of its properties including a surgery formula. Via surgery, the invariant Q is equivalent to an invariant Q' of null-homologous knots in rational homology spheres, that is conjecturally equivalent to the two-loop part of the Kontsevich integral. We generalize the construction of Q' to obtain a topological construction for an invariant that is conjecturally equivalent to the whole Kricker rational lift of the Kontsevich integral for null-homologous knots in rational homology spheres.

math.GT↗
arXiv2010

On the cube of the equivariant linking pairing for knots and 3-manifolds of rank one

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M)/Torsion. This article is devoted to the study of the invariant Q. We prove many properties for this invariant including two surgery formulae. Via surgery, the invariant Q is equivalent to an invariant of null-homologous knots in rational homology spheres, that coincides with the two-loop part of the Kricker rational lift of the Kontsevich integral, at least for knots with trivial Alexander polynomial in integral homology spheres.

math.GT↗
arXiv2007

Surgery formulae for finite type invariants of rational homology 3--spheres

We first present three graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the form $$\sum_{I \subset N} (-1)^{\sharp I}Z_n(M_I)$$ where $N$ is the set of components of a framed algebraically split link $L$ in a rational homology sphere $M$, and $M_I$ denotes the manifold resulting from the Dehn surgeries on the components of $I$. The first formula treats the case when $L$ is a boundary link with $n$ components, while the second one is for $3n$--component algebraically split links. In the third formula, the link $L$ has $2n$ components and the Milnor triple linking numbers of its 3--component sublinks vanish. The presented formulae are then applied to the study of the variation of $Z_n$ under a $p/q$-surgery on a knot $K$. This variation is a degree $n$ polynomial in $q/p$ when the class of $q/p$ in $\QQ/\ZZ$ is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.

math.GT↗
arXiv2005

Clover calculus for homology 3-spheres via basic algebraic topology

We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to this filtration is generated by Jacobi diagrams. Here, we express elements associated to LP-surgeries as explicit combinations of these Jacobi diagrams in (G_n)_n. The obtained coefficient in front of a Jacobi diagram is computed like its weight system with respect to a Lie algebra equipped with a non-degenerate invariant bilinear form, where cup products in 3-manifolds play the role of the Lie bracket and the linking number replaces the invariant form. In particular, this article provides an algebraic version of the graphical clover calculus developed by Garoufalidis, Goussarov, Habiro and Polyak. This version induces splitting formulae for all finite type invariants of homology 3-spheres.

math.GT↗
arXiv2004

Splitting formulae for the Kontsevich-Kuperberg-Thurston invariant of rational homology 3-spheres

M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We discuss the behaviour of Z under rational homology handlebodies replacements. The explicit formulae that we present generalize a sum formula obtained by the author for the Casson-Walker invariant in 1994. They allow us to identify the degree one term of Z with the Walker invariant for rational homology spheres.

math.GT↗
arXiv2004

On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rational homology 3-spheres

M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We review the Kontsevich-Kuperberg-Thurston construction and we provide detailed and elementary proofs for the invariance of Z. This article is the preliminary part of a work that aims to prove splitting formulae for this powerful invariant of rational homology spheres. It contains the needed background for the proof that will appear in the second part.

math.GT↗
arXiv2002

On configuration space integrals for links

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.

math.GT↗
arXiv2000

About the uniqueness and the denominators of the Kontsevich Integral

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel anomaly (that groups the Bott and Taubes anomalous terms) is a combination of diagrams with two univalent vertices; and we explicitly define the isomorphism of the space of Feynman diagrams which transforms the Kontsevich Integral into the Poirier limit of the perturbative expression of the Chern-Simons theory, as a function of the anomaly. We use this corollary to improve the Le estimates on the denominators of the Kontsevich Integral.

math.GT↗