Swatee Naik, Theodore Stanford
Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander polynomial if and only if it can be undone by doubled-delta moves.
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We show that the structure of a fibered knot, as a fiber bundle, is reflected in its knot quandle. As an application, we discuss finiteness and equivalence of knot quandles of concrete fibered 2-knots.
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 of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.
We show that any two bridge positions of a handlebody-knot are stably equivalent.
The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the following facts. For the ribbon-move on 2-knots, which is a kind of local-move on knots, we have the following: There is a ribbon-move-unknotting-number-two 2-knot. The ribbon-move-unknotting-number of 2-knots is unbounded. For the pass-move on 1-knots, which is a kind of local-move on…
We show several relations between local moves on 1-dimensional knots and those on high dimensional knots related by products of knots.
We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.
The $ν^+$-equivalence classes of genus one knots
本文研究纽结协调群上的 $ν^+$-等价，证明任意亏格为1的纽结必 $ν^+$-等价于三叶结、其镜像或平凡纽结。
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition…
This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let $K$ and $K'$ be 2-knots. Let $K$ and $K'$ are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimensional knot whose Seifert hypersurface is a punctured homology sphere.
Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants $τ$, $\varepsilon$, and $Υ$. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants defined by Kim and Livingston to show that these relations do not extend to stable equivalence.
R-equivalence classes of $\mathrm{Rot} \mathbb{E}^{2}$-colorings of torus knots
本研究引入纽结图广群着色的R等价关系，并确定特定条件下环面纽结图的$\u0001$mathrm{Rot} \u0001$mathbb{E}^{2}$着色R等价类。
A braidoid equivalence for spherical knotoids
本研究针对球面纽结类的辫状表示问题，通过细化辫状图的L-等价概念，建立了S²中（多）纽结类图以带标记辫状图闭包表示时的等价定理，完善了辫状理论与纽结类理论的对应关系。
Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Go…
Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S', K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope.
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 $F_{4}$) .
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 one must in general use $Ω_3$-moves of at least two different classes. To show this, knot diagram invariants that jump only under $Ω_3$-moves are introduced. Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, …
Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using marked graph diagrams.
Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using diagrams in ${\mathbb{R}}^3$.
We consider 32 homotopy classifications of knot projections (images of generic immersions from a circle into a 2-sphere). These 32 equivalence relations are obtained based on which moves are forbidden among the five type of Reidemeister moves. We show that 32 cases contain 20 non-trivial cases that are mutually different. To complete the proof, we obtain new tools, i.e., new invariants.
In this survey paper we present the $L$--moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3--manifolds and in handlebodies, as well as for virtual knots, for flat virtuals, for welded knots and for singular knots. The $L$--moves are local and they provide a uniform ground for formulating and proving braid equivalence theorems for any diagrammatic setting where the notion of braid and diagrammatic isotopy is defined, the statements…
Arc shift move and region arc shift move for twisted knots
本文提出了一种新的扭结无结操作——区域弧移操作，并研究了其在扭结简化中的应用与界限。
For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ and $D'$ necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in $4$-space.
Stably exotic knotted surfaces up to equivalence
该研究证明了4维流形中存在一类稳定奇异纽结曲面，其拓扑等价但经任意次内部、外部稳定化后仍不光滑等价，丰富了纽结曲面的等价性理论。
K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of $C_k$-move. In this paper we give another geometric condition in terms of Brunnian local move. The proof is simple and self-contained.
In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtual knot diagram can be turned into trivial knot using arc shift (region arc shift) moves. Based upon the arc shift move and region arc shift move, we define two virtual knot invariants, arc shift number and region arc shift number respectively.
The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links.
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.
We succeed to generalize spun knots of classical 1-knots to the virtual 1-knot case by using the `spinning construction'. That, is, we prove the following: Let $Q$ be a spun knot of a virtual 1-knot $K$ by our method. The embedding type $Q$ in $S^4$ depends only on $K$. Furthermore we prove the following: The submanifolds, $Q$ and the embedded torus made from $K,$ defined by Satoh's method, in $S^4$ are isotopic. We succeed to generalize the above construction to the virtual 2-knot case. Note that Satoh's method says nothing about the virtual 2-knot case. Rourke's interpretation of Satoh's met…
Virtualized Delta moves for virtual knots and links
该研究提出虚拟纽结与链环的虚拟化 Δ 移动，证明其对虚拟纽结是解结操作，并刻画了虚拟链环在该移动下等价的充要条件。
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.
Let $D(p,q)$ be the usual knot diagram of the $(p,q)$-torus knot, that is, $D(p,q)$ is the closure of the $p$-braid $(σ_1^{-1} σ_2^{-1}... σ_{p-1}^{-1})^q$. As is well-known, $D(p,q)$ and $D(q,p)$ represent the same knot. It is shown that $D(n+1,n)$ can be deformed to $D(n,n+1)$ by a sequence of $\{(n-1)n(2n-1)/6 \} + 1$ Reidemeister moves, which consists of a single RI move and $(n-1)n(2n-1)/6$ RIII moves. Using cowrithe, we show that this sequence is minimal over all sequences which bring $D(n+1,n)$ to $D(n,n+1)$.
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 all of the move types.
The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram
We propose some natural generalizations of Reidemeister moves that do not increase the number of crossings in the generated diagrams. Experimentations make us conjecture that this class of monotonic moves is complete for computing canonical forms and then deciding isotopy.
Given a uniformly convergent sequence of ambient isotopies $(H_n)_{n\in\mathbb{N}}$, bijectivity of the limit function $H_\infty$ together with a minor compactness condition guarantees that $H_\infty$ is also an ambient isotopy. By offloading the uniform convergence hypothesis to a more diagrammatic condition, we obtain sufficient conditions for performing countably-many Reidemeister moves. We use this to construct examples of tame knots with countably-many crossings and discuss what distinguishes these from similar-looking wild curves.
In this paper, we prove that given two cubical links of dimension two in ${\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively.
One-two-way pass-move for knots and links
本文提出一种由通过移动和$\#$-移动混合而成的局部移动$1$-$2$-移动，证明其对纽结的等价分类与通过移动一致，但所需移动次数不同。
Planar Equivalence of Knotoids and Quandle Invariants
针对平面纽结类分类难的问题，本文用受限广群着色与上循环不变量研究其等价性，还提出更强的三角广群上循环不变量，可用于检测平面性及分析对称性。
In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by a cyclic permutation of contiguous vertices of the $\mathbb{Z}^3$-lattice (with some restrictions), to describe some of the classic invariants and properties of the knots in terms of such cyclic permutations, by projecting onto a plane such that it is injective when restricted t…
This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diagram D' with the minimal possible number of crossings for the isotopy class of the knot represented by D. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain Z1, Z2, Z3 generalizing the classical Reidemeister moves R…
We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex.
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。
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…
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 than out of crossings. We describe an equivalence relation on clasp diagrams which produces the isotopy classes of knots as equivalence classes. This equivalence relation is generated by local moves similar to the Reidemeister moves. Clasp diagrams produce particularly simple Seifert surfaces for knots and lead to an explicit formula for the Alexander-Conway p…
If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need no merge operations to deform a rectangular diagram of the trivial knot to one with no crossings.
Virtualized Delta, sharp, and pass moves for oriented virtual knots and links
本文研究定向虚拟链环上的虚拟化 Delta、sharp 与 pass 移动，给出链环经这些局部移动等价的充要条件，并确立可达到的解结数下界。
Ma-Qiu index, presentation distance, and local moves in knot theory
本文通过马-邱指数研究结论中的局部移动与呈现距离的关系，提供了解结数的统一证明方法。
Writhes and $2k$-moves for virtual knots
本文研究了虚拟结的 $2k$-移动，证明了通过该移动相关的虚拟结的 $n$-绞拧数和奇绞拧数分别满足模 $k$ 和模 $2k$ 的同余关系，并给出了奇绞拧数模 $2k$ 同余类相同的充要条件。
A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched covering surface-knot by an addition of 1-handles with chart loops to a form such that its chart is the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications for the case when branched covering surface-knots have a non-zero number of bra…
Algebraic realizability of knot Floer-like complexes
该研究聚焦类纽结复形局部等价类可实现性的代数障碍，完成了$\mathbb{F}[U,V]$上相关等价类的分类，解决了Dai等人提出的问题。
Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classical knots. However, many virtual knots survive after this simplification. We construct invariants of these objects and present their applications to minimality problems of virtual knots as well as some questions related to graph-links. One can easily generalize these results for …
S-Equivalence of Band-Twisted Genus One Knots
本文构造了一列亏格一结，它们彼此 $S$-等价，但可通过 Jones 多项式区分，回应了 Kirby 问题列表中的问题 1.6 (K3)。
We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.
We use the terms, knot product and local move, as defined in the text of the paper. Let $n$ be an integer$\geqq3$. Let $\mathcal S_n$ be the set of simple spherical $n$-knots in $S^{n+2}$. Let $m$ be an integer$\geqq4$. We prove that the map $j:\mathcal S_{2m}\to\mathcal S_{2m+4}$ is bijective, where $j(K)=K\otimes$Hopf, and Hopf denotes the Hopf link. Let $J$ and $K$ be 1-links in $S^3$. Suppose that $J$ is obtained from $K$ by a single pass-move, which is a local-move on 1-links. Let $k$ be a positive integer. Let $P\otimes^kQ$ denote the knot product $P\otimes\underbrace{Q\otimes...\otimes …
On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots
该研究解决了吉川定向第五变换不可被其他变换替代的开放问题，扩展了浸入曲面链的代数工具包并开发了双quandle着色理论，还证明了无限多对群同构但结quandle不同的带边2-结。
Let $n$ be an integer$\geqq0$. Let $S^{n+2}_1$ (respectively, $S^{n+2}_2$) be the $(n+2)$-sphere embedded in the $(n+4)$-sphere $S^{n+4}$. Let $S^{n+2}_1$ and $S^{n+2}_2$ intersect transversely. Suppose that the smooth submanifold, $S^{n+2}_1 \cap S^{n+2}_2$ in $S^{n+2}_i$ is PL homeomophic to the $n$-sphere. Then $S^{n+2}_1$ and $S^{n+2}_2$ in $S^{n+2}_i$ is an $n$-knot $K_i$. We say that the pair $(K_1,K_2)$ of n-knots is realizable. We consider the following problem in this paper. Let $A_1$ and $A_2$ be n-knots. Is the pair $(A_1,A_2)$ of $n$-knots realizable? We give a complete characteriz…
Isotopy and equivalence of knots in 3-manifolds
本文研究了3流形中结的同伦与等价性，证明了在素、闭合、定向的3流形中，等价结当且仅当保持定向的映射类群为平凡群。对于不可约的闭合定向3流形，作者证明了保持自由同伦类的定向保形映射与恒等映射同伦。在S¹×S²的情况下，给出了许多结的同伦类被Gluck扭变的例子。
In this paper, we introduce an equivalence relation on the set of local moves and classify local moves, called the extended $ST$-moves, up to the equivalence. Moreover, by inducing a binary relation on the set of equivalence classes of local moves, we show that an extended $ST$-move realizes the crossing change or the $SH(2)$-move. In addition, for any oriented knot and two extended $ST$-moves, we disscus the magnitude relation between the unknotting numbers of the knot via the moves, and show that there is an extended $ST$-move except $SH$-moves so that the knot can be transformed into the tr…
A new unknotting operation for classical and welded links
本文提出一种新的无结操作，能够将任何结或链简化为无结或无链，通过有限的无结操作和 Reidemeister 操作，并分析无结操作下的结距离。
Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual homological notion of linking by taking into account the fundamental group of the ambient manifold and often map onto infinitely generated groups. The knot invariants generalize the cyclic (type 1) invariants of Kirk and Livingston and when taken with respect to certain preferred knots, called spherical knots, relative self-linking numbers are characterized geometrica…
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