Reidemeister Moves in Gauss Diagrams

Ordering the Reidemeister moves of a classical knot

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.

Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

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 type II for a pair of diagrams of the trivial virtual-knot. This implies that for any virtual-knot diagram $D$, there exists a diagram $D'$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister…

Unknotting number and number of Reidemeister moves needed for unlinking

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 respect to the number of crossings. Assuming a certain conjecture on unknotting numbers of a certain series of composites of torus knots, we show that the above diagrams need quadratic number of Reidemeister moves for being splitted.

需要Reidemeister二次移动数才能解开的解结图

Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle

我们提出了一个序列的图的unknot的最小数量的Reidemeister移动所需的通过平凡的图是二次的交叉点的数量。这些界限适用于$S^2$和$\R^2$。

关于Reidemeister移动2简化平面浸入曲线的必要性

On the necessity of Reidemeister move 2 for simplifying immersed planar curves

在2001年，Oestlund证明Reidemeister的移动1和3足以描述一个同伦从任何一般的浸入到平面的圆的标准嵌入的圆。我们证明这个猜想是错误的。

旋转Reidemeister移动的最小生成集

Minimal generating sets of rotational Reidemeister moves

本文研究旋转缠绕图的Reidemeister移动生成集，证明无框架和框架情形最小生成集大小分别为8和9，并推论有向框架链环最小生成集含5个移动。

A polynomial upper bound on Reidemeister moves

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 Reidemeister moves required to transform a diagram of the link into a disconnected diagram.

Minimal generating sets of Reidemeister moves

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 introduce a minimal generating set of four oriented Reidemeister moves, which includes two O1 moves, one O2 move, and one O3 move. We then study which other sets of up to 5 oriented moves generate all moves, and show that only few of them do. Some commonly considered sets are shown not…

每一个Reidemeister移动需要每一个结类型

Every Reidemeister move is needed for each knot type

我们表明，每一个结类型承认一对图，不能不使用Reidemeister Ω_2-移动相同。我们还表明，我们的证明是兼容的其他移动类型的已知结果，在这个意义上说，每一个结类型承认一对图，不能不使用所有的移动类型相同。

OU数与III型Reidemeister移动对链接图的性质

The OU number and Reidemeister moves of type III for link diagrams

本文提出OU数和非自OU序列，用于确定链接图之间III型Reidemeister移动的下界。

Realizing Exterior Cromwell moves on rectangular diagrams by Reidemeister moves

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 upper bound for the number of Reidemeister moves needed for unknotting a knot diagram of the trivial knot. However, exchange or merge moves on the top and bottom pairs of edges of rectangular diagrams are not considered in the proof. In this paper, we show that there is a rectangul…

Irregular smoothing and the number of Reidemeister moves

In the previous paper, we considered a link diagram invariant of Hass and Nowik type using regular smoothing and unknotting number, to estimate the number of Reidemeister moves needed for unlinking. In this paper, we introduce a new link diagram invariant using irregular smoothing, and give an example of a knot diagram of the unknot for which the new invariant gives a better estimation than the old one.

Minimal generating sets of directed oriented Reidemeister moves

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 $8$ directed Polyak moves $\{ Ω{1a}^\uparrow, Ω{1a}^\downarrow, Ω{1b}^\uparrow, Ω{1b}^\downarrow, Ω{2a}^\uparrow, Ω{2a}^\downarrow, Ω{3a}^\uparrow, Ω{3a}^\downarrow \}$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing…

有向虚拟纽结的Reidemeister移动的生成集

A generating set of Reidemeister moves of oriented virtual knots

针对有向虚拟纽结Reidemeister移动生成集研究不足的问题，研究枚举得到17种去冗余的有向虚拟Reidemeister移动，证明4元素子集为其生成集，简化了有向虚拟纽结不变量的验证工作。

纽结图的不变量与Reidemeister移动之间的关系

Invariants of knot diagrams and relations among Reidemeister moves

本文介绍了一种最精细的雷德迈斯特动作分类法。特别是，这种分类区分了一些$Ω_3$ -移动，它们的区别仅在于移动中涉及的三条链在结上的排序方式。为了将同位素结的结图相互转换，通常必须使用$Ω_3$ -至少两个不同类的移动。为了说明这一点，引入了仅在$Ω_3$ -移动下跳跃的结图不变量。同位素纽结的纽结图可以通过一系列里德迈斯特移动连接，在总共24个类别中只有6个类别。这一结果可应用于纽结理论中，简化数学纽结不变量不变性的证明。特别地，给出了高斯图上的函数定义纽结不变量的一个判据。

关于关于Reidemeister move I和II极小的链环图

On link diagrams that are minimal with respect to Reidemeister moves I and II

本文定义了关于Reidemeister move I和II极小的链环图，证明了无平凡分离分支链环的极小图及其交叉点数唯一，并探讨了Reidemeister move III对极小图的作用。

Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves

Arnold introduced invariants $J^+$, $J^-$ and $St$ for generic planar curves. It is known that both $J^+ /2 + St$ and $J^- /2 + St$ are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves for uknotting. $J^- /2 + St$ works well for unmatched RII moves. However, it works only by halves for RI moves. Let $w$ denote the writhe for a knot diagram. We show that $J^- /2 + St \pm w/2$ works well also for RI moves, and demonstrate that it gives a precise estimation for a certain knot d…

On Type II Reidemeister moves of links

Östlund (2001) showed that all planar isotopy invariants of generic plane curves that are unchanged under cusp moves and triple point moves, and of finite degree (in self-tangency moves) are trivial. Here the term "of finite degree" means Arnold-Vassiliev type. It implies the conjecture, which was often called Östlund conjecture: "Types I and III Reidemeister moves are sufficient to describe a homotopy from any generic immersion from the circle into the plain to the standard embedding of the circle". Although counterexamples are known nowadays, there had been no (easy computable) function that…

Reidemeister Moves and Groups

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev, who thought all free knots to be trivial). As it turned out, these new objects are highly non-trivial, and even admit non-trivial cobordism classes. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: an element has its minimal representative which "l…

奇异Reidemeister移动的最小生成集及其分类

Minimal Generating Sets of Singular Reidemeister Moves and Their Classification

通过引入不变量，完全解决了定向奇异Reidemeister移动的最小生成集问题，发现恰好有96个包含极小生成集，并确定了无向情况下的8个生成集。

关于每个结类型的Reidemeister移动的多项式上界

A polynomial upper bound on Reidemeister moves for each link type

本文研究了每个结类型的Reidemeister移动的多项式上界，并展示了如何通过显式计算多项式来确定结的识别问题复杂性。

Automated Reidemeister Moves: A Numerical Approach to the Unknotting Problem

In mathematics, a knot is a single strand of string crossed over itself any number of times, and connected at the ends. The Reidemeister Moves have been proven to be the three core moves necessary to fully untangle a knot. Some knots can be untangled to a loop (the unknot), while others are fundamentally knotted. We define four generalized moves based on the Reidemeister Moves. These moves have the capability of untangling knots which must be made more complicated before they can be simplified. With these moves, we construct a computer program which reads the two-dimensional projection of a kn…

解开打结所需的Reidemeister移动次数

The number of Reidemeister Moves Needed for Unknotting

存在正常数$c_1$，使得对于表示解结的任何图$D$，存在将其转换为平凡结图的至多$2^{c_1 n}$ Reidemeister移动的序列，$n$是$D$中的交叉数。类似的结果也适用于嵌入在紧的、可定向的、三角化的$PL$ 3-流形$M$内部的1-骨架中的多边形结$K$上的初等移动。存在正的常数$c_2$，使得对于每个$t \geq 1$，如果$M$由$t$四面体组成，并且$K$是不打结的，则在$M$中存在至多$2^{c_2 t}$基本移动的序列，其将$K$变换为包含在$M$的一个四面体内的三角形。我们得到明确的值为$c_1$和$c_2$。

凸曲面投影的Legendrian Reidemeister移动

Legendrian Reidemeister moves for the convex surface projection

本文证明了加厚凸曲面中Legendrian结的Legendrian Reidemeister定理，为凸曲面投影提供了完整的移动等价刻画。

Some chain maps on Khovanov complexes and Reidemeister moves

We introduce some chain maps between Khovanov complexes. Each of the chain maps commutes with a chain homotopy map and a retraction maps which obtain a Reidemeister invariance of Khovanov homology.

Reidemeister 和 movie 移动用于 involution 链

Reidemeister and movie moves for involutive links

本文研究了 involution 链的等价性，通过建立等价的 movie 移动集合，证明了等价的 Reidemeister 定理，并利用嵌入的等价 Morse 理论分析了等价的 Reidemeister 移动的循环。

Reidemeister 移动的最小生成集

Minimal Generating sets of Reidemeister moves

研究确定了Reidemeister移动的最小生成集，分类了包含一致型II移动和辫型III移动的生成集，并发现16个候选集合中12个被证明是最小的，其余4个仍待验证。

通过限制的 Reidemeister 移动证明交叉数的可加性

Additivity of Crossing Number via Restricted Reidemeister Moves

本文通过限制的 Reidemeister 移动证明了交叉数的可加性，并探讨了其拓扑意义。

A Reidemeister type theorem for petal diagrams of knots

We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing exchanges, which do not change the isotopy class of the underlying knot. We prove that any two permutations which represent isotopic knots can be related by a sequence of these moves and their inverses.

On Reidemeister invariance of the Khovanov homology group of the Jones polynomial

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 of Khovanov homology groups. In this paper, homotopy maps are obtained explicitly for the other Reidemeister moves, i.e. second and third.

On elementary moves of singular Legendrian knots

We have two results. First, we give 96 generating sets oriented singular Reidemeister moves; it is an answer to a question by Bataineh, Khaled, Elhamdadi, and Hajij who give a generating set of oriented singular Reidemeister moves using their computation. Second, in the theory of plane curve and Legendrian knots introduced by V. I. Arnold, we select which moves survive as those of Legendrian singular knots and fronts diagrammatically and explicitly.

Generating sets of Reidemeister moves of oriented singular links and quandles

We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.

Shadow movies not arising from knots

A shadow diagram is a knot diagram with under-over information omitted; a shadow movie is a sequence of shadow diagrams related by shadow Reidemeister moves. We show that not every shadow movie arises as the shadow of a Reidemeister movie, meaning a sequence of classical knot diagrams related by classical Reidemeister moves. This means that in Kaufman's theory of virtual knots, virtual crossings cannot simply be viewed as classical crossings where which strand is over has been left `to be determined'.

关于自由结和自由链

On Free Knots and Links

经典结和虚结都是以形式高斯图模对应于Reidemeister移动的一些抽象移动而出现的。如果我们忘记上/下交叉结构，并将结的数量以相同的Reidemeister移动为模，我们会得到虚拟结的戏剧性简化，这会杀死所有经典的结。然而，许多虚拟结在这种简化之后仍然存在。我们构造了这些对象的不变量，并将其应用于虚结点的极小性问题以及与图链接相关的一些问题。人们可以很容易地将这些结果推广到可定向的情况，并将其应用于解决不可逆性问题。这些不变量背后的主要思想是一些几何结构，它将一般等价简化为仅模Reidemeister - 2移动的等价。

Unknotting twisted knots with Gauss diagram forbidden moves

Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two "forbidden moves" $F_{1}$ and $F_{2}$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three "forbidden moves" $T_{4}$, $F_{1}$ (or $F_{2}$) and $F_{3}$ (or $F_{4}$) .

Drawings of Kn with the same rotation scheme are the same up to Reidemeister moves. Gioan's Theorem

A {\em good drawing\/} of $K_n$ is a drawing of the complete graph with $n$ vertices in the sphere such that: no two edges with a common end cross; no two edges cross more than once; and no three edges all cross at the same point. Gioan's Theorem asserts that any two good drawings of $K_n$ that have the same rotations of incident edges at every vertex are equivalent up to Reidemeister moves. At the time of preparation, 10 years had passed between the statement in the WG 2005 conference proceedings and our interest in the proposition. Shortly after we completed our preprint, Gioan independently…

Cube diagrams and 3-dimensional Reidemeister-like moves for knots

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 homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.

Strong and weak (1, 3) homotopies on knot Projections

Strong and weak (1, 3) homotopies are equivalence relations on knot projections, defined by the first flat Reidemeister move and each of two different types of the third flat Reidemeister moves. In this paper, we introduce the cross chord number that is the minimal number of double points of chords of a chord diagram. Cross chord numbers induce a strong (1, 3) invariant. We show that Hanaki's trivializing number is a weak (1, 3) invariant. We give a complete classification of knot projections having trivializing number two up to the first flat Reidemeister moves using cross chord numbers and t…

(1, 2) and weak (1, 3) homotopies on knot projections

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3).

Tangles, Generalized Reidemeister Moves, and Three-Dimensional Mirror Symmetry

Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch l…

An Index-Type Invariant of Knot Diagrams Giving Bounds for Unknotting Framed Unknots

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 significantly harder than unknotting using regular Reidemeister moves. We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under Ω3 moves depends only on the forward/backward character of the move, similar to how the change of …

Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space

The dual to a tetrahedron consists of a single vertex at which four edges and six faces are incident. Along each edge, three faces converge. A 2-foam is a compact topological space such that each point has a neighborhood homeomorphic to a neighborhood of that complex. Knotted foams in 4-dimensional space are to knotted surfaces, as knotted trivalent graphs are to classical knots. The diagram of a knotted foam consists of a generic projection into 4-space with crossing information indicated via a broken surface. In this paper, a finite set of moves to foams are presented that are analogous to t…

结和链接的自然移动

Natural Moves for Knots and Links

我们提出了一些自然的概括Reidemeister移动，不增加生成的图中的交叉点的数量。实验结果使我们猜想这类单调移动对于计算标准形进而判定合痕是完备的。

Virtual Knot Groups

For a knot diagram $K$, the classical knot group $π_1(K)$ is a free group modulo relations determined by Wirtinger-type relations on the classical crossings. The classical knot group is invariant under the Reidemeister moves. In this paper, we define a set of quotient groups associated to a knot diagram $K$. These quotient groups are invariant under the Reidemeister moves and the set includes the extended knot groups defined by Boden et al and Silver and Williams.

虚同伦

Virtual Homotopy

两个焊接（分别为虚拟）链接图是同伦的，如果一个可以转化为其他的一系列的扩展Reidemeister移动，经典Reidemeister移动，和自交叉的变化。在本文中，我们扩展Milnor的亩和酒吧亩不变量焊接和虚拟链接。最后，我们总结了本文的几个例子，并计算μ不变量使用的Magnus扩展和Polyak的绞关系的μ不变量。

H(n)-移动是虚拟和焊接链环的解结操作

The H(n)-move is an unknotting operation for virtual and welded links

本文将经典纽结的 H(n)-解结移动推广至虚拟与焊接链环，并证明虚拟化和禁止移动均可由有限次广义 Reidemeister 移动与 H(n)-移动实现。

Strong and weak (1, 2, 3) homotopies on knot projections

A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first non-trivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence …

Crossing indices, traits and the principle of indistinguishability

A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paired by a second Reidemeister move have the same index. We show that one can omit the second condition in the case of the universal index. As a consequence, we get the following principle of indistinguishability for classical knots: crossings of the same sign in a classical knot diagram can not be distinguished by any inherent property.

Links and Planar Diagram Codes

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code. PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan, et al. We present the set of PD-codes as a stand alone object and discuss its relationship with link diagrams. We give an explicit algorithm for reconstructing a knot diagram on a surface from a PD-code. We also discuss the intrinsic symmetries of PD-codes (i.e., invertibility and chirality). The moves analogous to the Reidemeister moves are also explored, and we show that the given set of PD-codes modulo th…

关于自由结

On Free Knots

我们证明，对于一些结一样的对象，可以很容易地识别非等价w.r.t.所有Reidemeister移动通过研究一些等价类模只有第二Reidemeister移动。有应用程序虚拟结，图形链接和循环图。

Quandle theory and the optimistic limits of the representations of link groups

When a boudnary-parabolic representation of a link group to PSL(2,$\mathbb{C}$) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related with the Reidemeister moves, so their method changes quite naturally under the Reidemeister moves. In this article, we apply their method to the potential function, which was used to define the optimsitic limit, and construct a saddle point of the function. This construction works f…

Ternary quasigroups in knot theory

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 show a degenerate subcomplex suitable for ternary quasigroups satisfying these axioms, and corresponding to the first Reidemeister move. For such ternary quasigroups with an additional condition that the primary operation equals to the second division operation, we also defin…

用高斯图禁忌步解开虚结

Unknotting virtual knots with Gauss diagram forbidden moves

禁止的移动可以与高斯图Reidemeister移动相结合，以获得移动序列，我们可以将任何高斯图（因此任何虚拟结）改变为任何其他，特别是包括未结图

Odd Khovanov Homology for Hyperplane Arrangements

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletion-restriction triples and dg-algebra structures. We also consider signed hyperplane arrangements, and generalize the odd Khovanov homology of Ozsvath-Rasmussen-Szabo from link projections to signed arrangements. We define hyperplane Reidemeister moves which generalize the usual R…