Related papers: A Stringy Generalization of the Kontsevich Integra…
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For…
We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic…
The globalization of Kontsevich's local formula (resp., the perturbative expansion of the Poisson sigma model) is described in down-to-earth terms.
We use Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality…
In this series of papers, we established that $Q^{\gamma_3}_{d=4}(P)$ is a coboundary in 4D (paper II arXiv:2409.12555), and we presented a series of experimental results about the (non)trivialisation of Kontsevich graph flows of…
A broader definition of generalized truncations of graphs is introduced followed by an exploration of some standard concepts and parameters with regard to generalized truncations.
We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a…
In this paper, we directly derive generalized mirror transformation of projective hypersurfaces up to degree 3 genus 0 Gromov-Witten invariants by comparing Kontsevich localization formula with residue integral representation of the virtual…
Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present paper we reprove their theorem using many of the same ideas, but with the key…
Every link in R^3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.
We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various…
The work provides a brief intuitive overview theory of graph on surfaces. We considers graphs with an additional structure, wich we call discs with ribbons, also known as one-vertex ribbon graphs. And solves the problem (Skopenkov's) about…
We show how Conway's multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel-Turaev, of the…
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov…
Regularization has become a primary tool for developing reliable estimators of the covariance matrix in high-dimensional settings. To curb the curse of dimensionality, numerous methods assume that the population covariance (or inverse…
We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph…
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…
We study generalizations of the Schr\"odinger problem in statistical mechanics in two directions: when the density is constrained at more than two times, and when the joint law of the initial and final positions for the particles is…
The integrality of the Kontsevich integral and perturbative invariants is discussed. We show that the denominator of the degree $n$ part of the Kontsevich integral of any knot or link is a divisor of $(2!3!... n!)^4(n+1)!$. We also show…
In this paper, we introduce \textit{graph-pretzel links}, a generalization of classical pretzel links based on spatial graph projections. As our main result, we investigate a subfamily associated with the complete graph on four vertices to…