arXiv research

The paper defines new homotopy relations on knot projections and classifies certain knot types.

Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.

We show that some ternary quasigroups appear naturally as invariants of classical links and links on surfaces. We also note how to obtain from them invariants of Yoshikawa moves. In our previous paper, we defined homology theory for algebras satisfying two axioms derived from the third Reidemeister move. In this paper,…

We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct a ternary invariant of curves immersed in compact surfaces, conside…

New sequences prove some link diagrams can't be transformed by specific moves.

New moves for singular knots identified and described.

As Oleg Viro describes in his paper, the most fundamental property of the Khovanov homology group is their invariance under Reidemeister moves. Viro constructes Khovanov complex and homology consisting of Jordan curves with sign and also gives a proof for the only case of first Reidemeister move by using his definition…

New homotopy types and invariants defined for knots.

Minimal sets of moves for isotopic knots and trivalent graphs identified.

Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.

We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We also introduce singular semiquandles and virtual singular semiquandles which define…

Minimal generating sets of Reidemeister moves identified and classified.

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodl…

Minimal sets of moves for rotational Reidemeister diagrams are identified.

A new type of knot energy is presented via real life experiments involving a thin resilient metallic tube. Knotted in different ways, the device mechanically acquires a uniquely determined (up to isometry) normal form at least when the original knot diagram has a small number of crossings, thus outperforming the famous…

Calculates lower bounds for type III Reidemeister moves in link diagrams.

The paper introduces a semiquandle for flat virtual knots and connects it to u-polynomials.

We provide an upper bound on the number of ordered Reidemeister moves required to pass between two diagrams of the same link. This bound is in terms of the number of unordered Reidemeister moves required.

Paper solves the minimal generating set problem for singular Reidemeister moves.

New moves prove crossing number sum for knots.

In this note we present a short proof that the 4 oriented Reidemeister moves of type 2 together with any one of the 8 oriented Reidemeister moves of type 3 are sufficient to imply the other 7.

We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of a quandle cocycle invariant, we determine the necessity of Reidemeister moves of…

This study simplifies verification of invariants in oriented virtual knots.

New groups defined from knot diagrams, invariant under Reidemeister moves.

Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being unknotted with resp…

Study shows that splitting links requires an arbitrarily large number of extra crossings.

The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the O1 and O2 moves, and 8 versions of the O3 move. We introd…

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.

New moves help untangle complex knots.

In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to D is unique. The proof al…

Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $…

32 knot projections classified based on forbidden Reidemeister moves.

Polynomial bound on Reidemeister moves for each link type.

The H(n)-move simplifies virtual and welded knots and links.

We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are sorted by type.

Study of equivariant movie moves for involutive links.

We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar theorem for split links, which provides a polynomial upper bound on the number of …

We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links.

Study Alexander polynomials of links in 3-torus.

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

We show that every knot type admits a pair of diagrams that cannot be made identical without using Reidemeister Omega_2-moves. We also show that our proof is compatible with known results for the other move types, in the sense that every knot type admits a pair of diagrams that cannot be made identical without using al…

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 and looped graphs.

If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by I. A. Dynnikov. Using this, Henrich and Kauffman gave an up…

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some $Ω_3$-moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other …

Three hard diagrams of the unknot require extra crossings to simplify.

Graphoids are topological invariants of virtual graph diagrams.

Paper develops invariants for spherical curves using chord diagrams.