Related papers: On a theorem of Kontsevich
We discuss the cohomology of the bridgeless graph complex, that is, the subcomplex of the Kontsevich graph complex spanned by bridgeless graphs.
In this paper, we develop a diamond graph theory and apply the theory to the (co)homology of the Lie algebra generated by positive systems of the classical semi-simple Lie algebras over the field of complex numbers. As an application, we…
We consider graph complexes with a flow and compute their cohomology. More specifically, we prove that for a PROP generated by a Koszul dioperad, the corresponding graph complex gives a minimal model of the PROP. We also give another proof…
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…
We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of…
Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of…
Kontsevich's graphs allow encoding multi-vectors whose coefficients are differential-polynomial in the coefficients of a given Poisson bracket on an affine real manifold. Encoding formulas by directed graphs adapts to the class of…
An analog of Kreimer's coproduct from renormalization of Feynman integrals in quantum field theory, endows an analog of Kontsevich's graph complex with a dg-coalgebra structure. The graph complex is generated by orientation classes of…
J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case…
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)…
We prove that the inclusion from oriented graph complex into graph complex with at least one source is a quasi-isomorphism, showing that homology of the "sourced" graph complex is also equal to the homology of standard Kontsevich's graph…
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…
In the paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots in R^d, d>=3. The first term of the Vassiliev spectral…
We introduce and study the notion of a dual Feynman transform of a modular operad. This generalizes and gives a conceptual explanation of Kontsevich's dual construction producing graph cohomology classes from a contractible differential…
An A-infinity algebra is a generalization of a associative algebra, and an L-infinity algebra is a generalization of a Lie algebra. In this paper, we show that an L-infinity algebra with an invariant inner product determines a cycle in the…
We study Maxim Kontsevich's graph complex $GC_d$ for any integer $d$ as well as its oriented and targeted versions, and show new short proofs of the theorems due to Thomas Willwacher and Marko Zivkovic which establish isomorphisms of their…
We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a…
We follow the same technics we used before in \cite{AZ} of extending knot Floer homology to embedded graphs in a 3-manifold, by using the Kauffman topological invariant of embedded graphs by associating family of links and knots to a such…
We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as…
We define a Courant bracket on an associative algebra using the theory of Hochschild homology, and we introduce the notion of Dirac algebra. We show that the bracket of an omni-Lie algebra is quite a kind of Courant bracket.