Related papers: Partial duality and Bollobas and Riordan's ribbon …
We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are…
We study a quantum version of the $n$-dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial…
The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded…
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs…
Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this…
We investigate combinatorial properties of a graph polynomial indexed by half-edges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We…
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…
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While…
In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…
We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed…
We introduce the notion of a Hodge-Riemann pair of cohomology classes that generalizes the classical Hodge-Riemann bilinear relations, and the notion of a Bogomolov pair of cohomology classes that generalizes the Bogomolov inequality for…
We generalize the construction of Akimova and Manturov, define the label bracket for knotted trivalent graphs in $\mathbb{R}^3$ and show it defines an isotopy invariant of such graphs.
In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.
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…
In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalized knotoids to allow arbitrarily…
In the present paper, we study a relation between the cohomology of moduli stacks of smooth and proper curves $\mathcal M_{g,n}$ and the cohomology of ribbon graph complexes. The main results of this work are proofs of T. Willwacher's…
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some…
In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and…
The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial…
The study of Feynman rules is much facilitated by the two Symanzik polynomials, homogeneous polynomials based on edge variables for a given Feynman graph. We review here the role of a recently discovered third graph polynomial based on…