相关论文: Sub-chord diagrams of knot projections

This paper demonstrates an approach for developing a framework to produce invariants of base-point-free generic spherical curves under some chosen local moves from Reidemeister moves using based chord diagrams. Our invariants not only…

A triple chord is a sub-diagram of a chord diagram that consists of a circle and finitely many chords connecting the preimages for every double point on a spherical curve, and it has exactly three chords giving the triple intersection. This…

A generic immersion of a circle into a $2$-sphere is often studied as a projection of a knot; it is called a knot projection. A chord diagram is a configuration of paired points on a circle; traditionally, the two points of each pair are…

We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves $S^1 \to R^2$. They are a direct analog of similar techniques for knot invariants and are based on the calculus of {\em…

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a…

The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which…

Every knot projection is simplified to the trivial spherical curve not increasing double points by using deformations of types 1, 2, and 3 which are analogies of Reidemeister moves of types 1, 2, and 3 on knot diagrams. We introduce RII…

We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather…

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some $\Omega_3$-moves that only differ in how the three strands that are involved in the move…

Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to…

A chord diagram refers to a set of chords with distinct endpoints on a circle. The intersection graph of a chord diagram $\cal C$ is defined by substituting the chords of $\cal C$ with vertices and by adding edges between two vertices…

In this paper, we give a definition of $\mathbb{Z}$-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by $\sum_i \alpha_i x_i$. Then, we introduce certain elements of the free…

The Reidemeister theorem states that any link in $3$-space can be encoded by a diagram (a suitably decorated projection) on a plane, and provides a finite set of combinatorial moves relating two diagrams of the same link up to isotopy. In…

We provide a way to produce knots in $S^3$ from signed chord diagrams, and prove that every knot can be produced in this way. Using these diagrams, we generalize the fundamental theorem of finite type invariants. We also provide moves for…

We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister…

Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord…

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general…

In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot…

In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link…

We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is…