arXiv research

Solves Skopenkov's problem on graph embedding criteria.

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.

Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual link diagrams, that are represented by the same ribbon graphs are related to eac…

Paper studies metric ribbon graphs and provides a recursion for their volumes.

Extends Heisenberg homology to ribbon graphs.

In this paper we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus the ribbon graph minor relation is incompatible wi…

New equivalence relation on ribbon graphs connects to virtual links.

In this paper, we analyze the Bollobás and Riordan polynomial $\mathcal{R}$ for ribbon graphs with half-ribbons introduced in [Combinatorics, Probability and Computing 31, 507-549, 2022]. We prove the universality property of a multivariate version of $\mathcal{R}$ whereas $\mathcal{R}$ itself turns out to be universal…

We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollobás-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ri…

Explains partial duality for ribbon graphs in simple terms.

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram $L$, there is an associated ribbon graph whose quasi-trees correspond bijectively to …

New property ensures non-looseness of ribbon boundaries.

Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or equivalently, embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a ch…

We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.

This paper extends results of Hatcher and Vogtmann's work "Cerf Theory for Graphs" to ribbon graphs. Given an orientable, punctured and basepointed surface Sigma, we prove that the space of ribbon graphs that can be drawn in Sigma is filtered by simplicial complexes. The k-th simplicial complex is (k-1)-dimensional, (k…

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chor…

Constructs homologies for ribbon graphs to recover Penrose polynomials.

Counterexamples to a conjecture on ribbon graph genus changes were found and proven.

Paper categorifies a polynomial related to ribbon graphs.

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the r…

In the present work we are going to give a formal exposition of the ribbon graphs topic based on notes of Labourie \cite{Lab}, since is difficult to find as such in the literature.

To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassil…

Paper generalizes pretzel links using spatial graphs.

Artin groups have finite stature based on vertex groups.

For a ribbon graph $G$ we consider an alternating link $L_G$ in the 3-manifold $G\times I$ represented as the product of the oriented surface $G$ and the unit interval $I$. We show that the Kauffman bracket $[L_G]$ is an evaluation of the recently introduced Bollobas-Riordan polynomial $R_G$. This results generalizes t…

A full Mealy automaton is associated with a graph and a square complex, which contains an anti-torus if and only if the automaton is bi-reversible and the graph is aperiodic.

New relations for Penrose polynomial at n=4 and n=3.

Let $n$ be a nonnegative integer, we use ribbon $n-$graph diagrams and the Yamada polynomial skein relations to construct an algebra ${\mathcal Y}_n$ which is shown to be closely related to the Temerley-Lieb Algebra. We prove that the algebra ${\mathcal Y}_2$ is isomorphic to some quotient of a three variables polynomi…

Graph neural networks have become one of the most important techniques to solve machine learning problems on graph-structured data. Recent work on vertex classification proposed deep and distributed learning models to achieve high performance and scalability. However, we find that the feature vectors of benchmark datas…

The Bollobás-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly-colored stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravi…

Uniform drift estimates found for random walks on graph products.

While convolutional neural networks (CNNs) have recently made great strides in supervised classification of data structured on a grid (e.g. images composed of pixel grids), in several interesting datasets, the relations between features can be better represented as a general graph instead of a regular grid. Although re…

Algorithm constructs triangulations for Heegaard splittings and related 3-manifolds.

Quantum model for knotted graphs from knot theory.

The Thistlethwaite theorem is extended to knotoids and linkoids.

It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a mar…

Quantum invariants for surfaces in 4D 2-handlebodies.

Paper describes a state sum formula for a graph coloring polynomial.

This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these tw…

Involutive Hopf monoids yield surface invariants.

Algorithm converts curves on ribbon surfaces to contact surgery diagrams.

Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.

A new approach to Morse theory using folded ribbon trees.

The paper finds 3-colorings of 2-sphere triangulations.

A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…

The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.

We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul an…