arXiv research

Graphs embeddable on torus and linklessly in 3D can be embedded linklessly in standard torus.

Researchers found two types of graphs for 6D torus manifolds with Euler number 6.

Proves existence of graph on torus with prescribed curvature.

Connected graph for twice-punctured torus curves.

The article proves $K5$ and $K_{3,3}$ are toroidal penny graphs.

Classification of torus homeomorphisms on fine curve graph completed.

Researchers found all embeddings of Kuratowski graphs on a double torus.

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-…

We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period. The proof of the main result is based on computing the Yamada skein algebra of the solid torus then proving that this algebra injects into the Kauffman bracket …

Abstract: Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph and have generalized rotation sets.

Angular measurements are often modeled as circular random variables, where there are natural circular analogues of moments, including correlation. Because a product of circles is a torus, a d-dimensional vector of circular random variables lies on a d-dimensional torus. For such vectors we present here a class of graph…

Embeddings of mapping tori for end-periodic graph maps are proven.

Classifies SU(2)-abelian graph manifolds with a single JSJ torus.

The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus $T^2 \subset \mathbb{R}^3$ and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecu…

To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a study of codimension~2 singularities of link diagrams. For closed braids with a fi…

To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral mo…

Study geodesics in 3-torus, determining complements' topology.

We compute Cayley graphs and automorphism groups for all finite $n$-quandles of two-bridge and torus knots and links, as well as torus links with an axis.

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links…

We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.

Hamiltonian cycles found in toroidal maps.

Optimal Euclidean structure minimizes energy in weighted toroidal graphs.

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can…

The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …

Discrete Green's functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green's functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for the cycle, torus, a…

In this paper we show that two Lagrangian graphs over the torus in $\mathbb{C}^n$ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in $\mathbb{C}^n$ ca…

Automorphisms of fine graphs for surfaces and tori are studied.

This paper is devoted to the study of special subgroups of the automorphism groups of Kronrod-Reeb graphs of a Morse functions on $2$-torus $T^2$ which arise from the action of diffeomorphisms preserving a given Morse function on $T^2$. In this paper we give a full description of such classes of groups.

Algorithms compute length spectra of torus graphs efficiently.

Counting tripods on a flat torus using lattice point counting.

The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.

The paper extends toric variety correspondence to 4D almost complex torus manifolds.

We study rerouting edges on surfaces without crossings.

The main objects of this paper are torus orbifolds that have exactly two fixed points. We study the equivariant topological type of these orbifolds and consider when we can use the results of the paper [DKS] (arXiv:1809.03678) to compute its integral equivariant cohomology, in terms of generators and relations, coming …

Study integrability of quantized six-vertex model on torus.

The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\mathcal{A}(Γ,α,\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(Γ,α,\nabla)$. Here, an $(m,n)$-type GKM …

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.

Geometrically, twist numbers on punctured tori are dense and non-continuous.

Study of SU(2,1) character varieties on one-holed torus.

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any…

We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$, $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\…

Study GKM actions on special manifolds with interval orbit spaces.

Lower bound on volumes of special mapping tori.

Classifies multigraphs for torus actions on 6D manifolds with isolated fixed points.

By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subco…

In this paper, we show how to construct graph theoretical models of n-dimensional continuous objects and manifolds. These models retain topological properties of their continuous counterparts. An LCL collection of n-cells in Euclidean space is introduced and investigated. If an LCL collection of n-cells is a cover of a…