arXiv research

Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…

Algorithm constructs Grushko decomposition of certain groups.

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.

Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split li…

Computes link invariants in real projective 3-space using topological vertex.

Characters from logarithmic VOAs linked to torus link invariants.

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations …

New invariants distinguish spatial graphs not previously possible.

Enumerating all 3-manifold triangulations of a given size is a difficult but increasingly important problem in computational topology. A key difficulty for enumeration algorithms is that most combinatorial triangulations must be discarded because they do not represent topological 3-manifolds. In this paper we show how …

We prove that the Cayley graph and the coset geometry of the von Dyck group $D(a,b,c)$ are linked by a vertex-to-edge duality.

Paper generalizes Yamada polynomial to virtual spatial graphs.

New formulas for spatial 2-bouquet graphs discovered.

New insights into non-torus links using topological vertices.

Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.

Proves minimal crossing diagrams for specific spatial graphs.

Classifies intrinsically linked tournaments by their score sequences.

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model…

We extend the construction of the DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type A from the iterated torus knots (our previous paper) to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is …

The Kauffman-Vogel polynomials are three variable polynomial invariants of $4$-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented $4$-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with $2$. Bataineh, Elha…

We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…

A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normal $d$-pseudomanifolds form a broader class than triangulations of connected closed $d$-manifolds for $d \geq 3$. Here, we classify all…

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…

We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine $W_L$…

Study uses knot theory to model RNA foldings, emphasizing both entanglement and intrachain interactions.

Biological and cellular systems are often modeled as graphs in which vertices represent objects of interest (genes, proteins, drugs) and edges represent relational ties among these objects (binds-to, interacts-with, regulates). This approach has been highly successful owing to the theory, methodology and software that …

The Bergman space and conformally flat 2-disk operads are linked to vertex operator algebras.

Enhances node embeddings in networks using topic modeling.

We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families, we verify a conjecture presented in Adams' "The Knot Book": If a vertex is remo…

We give an explicit construction of vertex-transitive tight triangulations of $d$-manifolds for $d\geq 2$. More explicitly, for each $d\geq 2$, we construct two $(d^2+5d+5)$-vertex neighborly triangulated $d$-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated …

Paper proves bounds on knot indices using bisected vertex leveling of plane graphs.

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projecti…

New proof shows unique symplectic fillings for certain surface singularity links.

New satellite constructions create infinite Brunnian links.

Network embedding aims to learn the low-dimensional representations of vertexes in a network, while structure and inherent properties of the network is preserved. Existing network embedding works primarily focus on preserving the microscopic structure, such as the first- and second-order proximity of vertexes, while th…

Virtual links were introduced by Kauffman in 1999. We characterize the virtual link invariants that are partition functions of vertex models (as considered by de la Harpe and Jones), both in the real and in the complex case. We show that for any fixed number of states, these invariants form an affine variety. Basic tec…

In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate feature maps for latent position graphs with positive definite link function $κ$, provided that the latent positions are i.i.d. from some distribution F. We then consider the exploitation task of vertex classi…

A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, t…

Paper proves ribbonlength grows linearly with knot complexity.

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be …

A ravel is a spatial graph which is non-planar but contains no non-trivial knots or links. We characterize when a Montesinos tangle can become a ravel as the result of vertex closure with and without replacing some number of crossings by vertices.

It is shown that if a real value PL-invariant of closed combinatorial manifolds admits a local formula that depends only on the f-vector of the link of each vertex, then the invariant must be a constant times the Euler characteristic.

We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first example is a geometric engineering of five-dimensional U(1) gauge theory with a ma…

DMTE integrates global connectivity for better text embeddings.

For any knot, a 3-sphere triangulation exists with a knotted edge.

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of $AC(S)$ coincides with the natural image of the exten…

Characterizes weakly linked pairs of complete graphs in 3D space.

This paper is a survey of some of the most elementary consequences of the JSJ-decomposition and geometrization for knot and link complements in the 3-sphere. Formulated in the language of graphs, the result is the construction of a bijective correspondence between the isotopy classes of links in $S^3$ and a class of ve…

Well-quasi-order proved for plane minors; polynomial-time algorithm for link diagrams.