arXiv research

New topological realization of Kontsevich graph complex for large dimensions.

In two seminal papers Kontsevich used a construction called_graph homology_ as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we s…

In this paper we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex which is generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology of an infinite-dimensional Lie algebra of symplectic vector fields.

We recall the construction of the Kontsevich graph orientation morphism $γ\mapsto {\rm O\vec{r}}(γ)$ which maps cocycles $γ$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(γ)(\mathcal{P})$ of Poisson bi-vectors on affine manifolds. We reveal in particular why there alw…

We introduce a "minimal" Kontsevich integral that generates the original Kontsevich integral while at the same time producing ribbons whose boundaries are the braids on which the minimal Kontsevich integral is evaluated. We generalize the definition of the Kontsevich integral to that of graphs in R^3 and study the beha…

We study the topology of a space parametrizing stable tropical curves of genus g with volume 1, showing that its reduced rational homology is canonically identified with both the top weight cohomology of M_g and also with the genus g part of the homology of Kontsevich's graph complex. Using a theorem of Willwacher rela…

We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step …

This is a substantially revised version. The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Eule…

Defines a new invariant from graph configurations in three-manifolds.

We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the…

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give…

The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function) with values in the symmetric algebra of the Lie algebra. Recently A. Kricker and …

Researchers determined the second homology group of a specific symplectic derivation Lie algebra.

We give a detailed explicit computation of weights of Kontsevich graphs which arise from connection and curvature terms within the globalization picture for the special case of symplectic manifolds. We will show how the weights for the curvature graphs can be explicitly expressed in terms of the hypergeometric function…

Critical graphs of quadratic differentials equidistribute in moduli space.

We analyze a functor from cyclic operads to chain complexes first considered by Getzler and Kapranov and also Markl. This functor is a generalization of the graph homology considered by Kontsevich, which was defined for the three operads Comm, Assoc, and Lie. More specifically we show that these chain complexes have a …

We prove that the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = \mathcal{Q}_{a:b} (\mathcal{P})$, the right-hand side of which is a linear combination of two differential monomials of degree four in a bi-vector $\mathcal{P}$ on an affine real Poisson manifold $N^n$, does infinitesimally preserve the space of Poisson…

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…

Modular operads are a special type of operad: in fact, they bear the same relationship to operads that graphs do to trees (i.e. simply connected graphs). One of the basic examples of a modular operad is the collection of Deligne-Mumford-Knudsen moduli spaces $\bar{M}_{g,n}$ of stable pointed algebraic curves; hence the…

We formulate a conjecture (already proven by A. Kricker) about the structure of Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent diagrams. We conjecture that these functions are rational functions of the exponentials …

We give a conceptual formulation of Kontsevich's `dual construction' producing graph cohomology classes from a differential graded Frobenius algebra with an odd scalar product. Our construction -- whilst equivalent to the original one -- is combinatorics-free and is based on the Batalin-Vilkovisky formalism, from which…

Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

$\newcommand{\poly}{_{\operatorname{poly}}^{\bullet}}\newcommand{\td}{(\operatorname{td}_{L/A}^{\nabla})^{\frac{1}{2}}}\newcommand{\cx}[1]{\operatorname{tot}\big(Γ(Λ^\bullet A^\vee)\otimes_R\mathcal{#1}\poly\big)}\newcommand{\cy}[1]{\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{#1}\poly)}$Kontsevich's formality the…

We introduce an algebra bundle of chord diagrams over the configuration space of N points in the complex plane on which we put the Knizhnik-Zamolodchikov connection. For that particular connection, the holonomy along a loop in the base is shown to be generating the Kontsevich integral for that loop's associated braid.

Study Hochschild cohomology of dg manifolds linked to integrable distributions.

Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.

Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.

New invariant counts graph configurations in 3D manifolds.

Graph potentials link to topological QFTs, with computational methods.

Formula connects linking coefficients to Kontsevich integral coefficients.

In this paper, we shall give an explicit Gauss diagram formula for the Kontsevich integral of links up to degree four. This practical formula enables us to actually compute the Kontsevich integral in a combinatorial way.

Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.

A map from 3-manifold skein to Lagrangian skein via holomorphic curve counting.

Embedding calculus invariants solve knot connectivity and grope cobordism problems.

We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral h…

We prove a homological version of a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.

The paper studies the n-loop Kontsevich invariant of knots with same Alexander polynomial.

The study finds arithmetic groups often in square-tiled surface monodromies.

Completed volumes match with combinatorial classes of the double ramification cycle.

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 an…

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le's theorem on the behaviour of the Kontsevich integral under cabling…

Kontsevich's classes distinguish smooth structures on fiber bundles.

This is an overview article on the Kontsevich integral written for the Encyclopedia of Mathematical Physics, to be published by Elsevier.

This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the pert…

The paper studies the index of a specific monodromy for origamis in a particular stratum.

Arithmetic Kontsevich-Zorich monodromy found in a specific origami surface.

As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree $k$ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k$. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and …