Related papers: Intersection graph and writhe polynomial

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from…

We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The…

We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the…

The intersection graph of a family of sets $\{S_{1},S_{2},\ldots,S_{n}\}$ is a graph whose vertex set is $\{S_{1},S_{2},\ldots,S_{n}\}$ and two distinct vertices are adjacent if the intersection of the corresponding sets is non-empty.…

This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way…

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with…

In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant…

In this note we give a new lower bound on the virtual crossing number via the writhe polynomial, which refines a result of B. Mellor. The proof is based on a new interpretation of the writhe polynomial. The characterization of the writhe…

We prove that for some knot-like objects one can easily recognize non-equivalence w.r.t. all Reidemeister moves by studying some equivalence classes modulo only 2nd Reidemeister moves. There are applications to virtual knots, graph-links…

Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via…

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a…

Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse…

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant…

This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave…

A meteor graph is a connected graph with no sources and sinks consisting of two disjoint cycles and the paths connecting these cycles. We prove that two meteor graphs are shift equivalent if and only if they are strongly shift equivalent,…

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads…

We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes…

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 construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by $\zeta$ meaning an analogy with $\zeta$-polynomial for virtual links. A degree of $\zeta$-polynomial estimates…

The present paper is a review of the current state of Graph-Link Theory (graph-links are also closely related to homotopy classes of looped interlacement graphs), dealing with a generalisation of knots obtained by translating the…