5-moves and Montesinos links

Volumes of Montesinos links

We show that the volume of any Montesinos link can be bounded above and below in terms of the combinatorics of its diagram. This was known for Montesinos links with at most two tangles, and those with at least five tangles. We complete the result for the remaining cases.

Computing twisted Alexander polynomials for Montesinos links

In recent years, twisted Alexander polynomial has been playing an important role in low-dimensional topology. For Montesinos links, we develop an efficient method to compute the twisted Alexander polynomial associated to any linear representation. In particular, formulas for multi-variable Alexander polynomials of these links can be easily derived.

A note on quasi-alternating Montesinos links

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we…

Quasi-alternating Montesinos links

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as provided bounds on unknotting number and slice genus, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.

Strongly quasipositive quasi-alternating links and Montesinos links

The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_0$ is alternating (where $L_0$ has the induced orientation), then $L$ is definite if and only if it is strongly quasipositive (up to mirroring). We also show that if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such t…

The fourth skein module and the Montesinos-Nakanishi conjecture for 3-algebraic links

We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a byproduct we prove the Montesinos-Nakanishi 3-move conjecture for 3-algebraic links (including 3-bridge links).

Montesinos链环的无迹SL(2,C)-表示

Tracefree ${\rm SL}(2,\mathbb{C})$-representations of Montesinos links

本文研究Montesinos链环的无迹SL(2,C)-表示，通过要求每个经线像的迹为零，完整确定了这类表示的共轭类。

蒙蒂西诺斯链环的环绕数

Linking numbers of Montesinos links

本研究针对蒙蒂西诺斯链环的环绕数计算问题，给出Tuler关于双分支有理链环绕数结论的简洁证明，提出有理链环绕数的数值算法，并借此得到蒙蒂西诺斯链环任意两分支的环绕数。

Doubly slice Montesinos links

This paper compares notions of double sliceness for links. The main result is to show that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end we prove a result classifying Seifert fibered spaces which admit a smooth embeddings into integer homology $S^1 \times S^3$s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links…

Montesinos纽结的Delta-解结数为一

Montesinos Knots with Delta-Unknotting Number One

本文研究$\Delta$-解结数为一的Montesinos纽结，提出其猜想性刻画，并给出不同$\Delta$-移动将纽结变为平凡纽结的实例。

Montesinos-Nakanishi 3-移动猜想：交叉数不超过20的链环

The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings

本文证明了交叉数不超过19（及除六个例外外的20交叉）链环满足Montesinos-Nakanishi 3-移动猜想，并完全分类了交叉数不超过20的链环，使用了包括Regina新代码在内的计算方法。

Mutation invariance of the arc index for some Montesinos knots

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

二桥链接和3-结蒙特斯林链接的 profinite 刚性

Profinite rigidity for two-bridge links and 3-tangle Montesinos links

本文证明了二桥链接和3-结蒙特斯林链接的补集基本群在紧致定向3流形的基本群中是profinite刚性的。

Kauffman-Harary conjecture holds for Montesinos Knots

The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman-Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman-Harary conjecture for all Montesinos links. Finall…

链接集上的一个偏序及其在对称联结中的应用

A preorder on the set of links with applications to symmetric unions

本文定义了链接的π- orbifold群，并探讨了链接之间的偏序关系，证明了特定条件下链接的结构特性，并应用该理论分析了对称联结的性质。

On The Jones Polynomial of Quasi-alternating Links

We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other than $(2,n)$-torus links, has no gaps. This would give a new property of quasi-alternating links and a simple obstruction criterion for a link to be quasi-alternating. We prove that the conjecture holds for quasi-alternating Montesinos links as well as quasi-alternating links with …

Burnside groups and $n$-moves for links

Let $n$ be a positive integer. M. K. Dabkowski and J. H. Przytycki introduced the $n$th Burnside group of links which is preserved by $n$-moves, and proved that for any odd prime $p$ there exist links which are not equivalent to trivial links up to $p$-moves by using their $p$th Burnside groups. This gives counterexamples for the Montesinos-Nakanishi $3$-move conjecture. In general, it is hard to distinguish $p$th Burnside groups of a given link and a trivial link. We give a necessary condition for which $p$th Burnside groups are isomorphic to those of trivial links. The necessary condition gi…

5-move equivalence classes of links and their algebraic invariants

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S^3 along a link. We use also the fact that a 5-move is a composition of two rational \pm (2,2)-moves (i.e. \pm 5/2-moves) and rational moves can be analyzed using the group of Fox co…

Extending Quasi-Alternating Links

Champanerkar and Kofman introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 cros…

Non-triviality of the Jones polynomial and the crossing numbers of amphicheiral knots

Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prove that there are infinitely many positive knots with no positive minimal crossing diagrams. Some relations to the twist number of a link, Mahler measure and the hyperbolic volume are given, for example explicit upper bounds on the volume for Montesinos and 3-braid links in terms …

三维流形的高次覆盖移动

Higher degree covering moves for 3-manifolds

本文为三维流形的分支覆盖提供高次覆盖移动的完整集合，证明稳定后仅需两个局部替换即可关联任意彩色链环，并给出d桥纽结覆盖为透镜空间的判定方法。

Milnor invariants, $2n$-moves and $V^{n}$-moves for welded string links

In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.

On C_n-moves for links

A C_n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a `higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to C_n-moves, under certain restrictions. Namely, we give a classification up to C_4-moves of 2-component links, 3-component Brunnian links and n-component C_3-trivial links, and we classify n-component link-homotopically trivial Brunnian links up to C_{n+1}-moves.

CF-moves for virtual links

Oikawa defined an unknotting operation on virtual knots, called a CF-move, and gave a classification of 2-component virtual links up to CF-moves by the virtual linking number and his $n$-invariant. In particular, it was proved that a CF-move characterizes the information contained in the virtual linking number for 2-component odd virtual links. In this paper, we extend this result by classifying odd virtual links and almost odd virtual links with arbitrary number of components up to CF-moves, using the virtual linking number. Moreover, we extend Oikawa's $n$-invariant and introduce two invaria…

Classification of links up to self $#$-move

A pass-move and a $#$-move are local moves on oriented links defined by L.H. Kauffman and H. Murakami respectively. Two links are self pass-equivalent (resp. self $#$-equivalent) if one can be deformed into the other by pass-moves (resp. $#$-moves), where non of them can occur between distinct components of the link. These relations are equivalence relations on ordered oriented links and stronger than link-homotopy defined by J. Milnor. We give two complete classifications of links with arbitrarily many components up to self pass-equivalence and up to self $#$-equivalence respectively. So our …

Classification of string links up to $2n$-moves and link-homotopy

Two string links are equivalent up to $2n$-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo $n$. Moreover, the set of the equivalence classes forms a finite group generated by elements of order $n$. The classification induces that if two string links are equivalent up to $2n$-moves for every $n>0$, then they are link-homotopic.

代数分离链环上的自delta移动与混合delta移动

Self and mixed delta-moves on algebraically split links

该研究探讨代数分离链环的自与混合delta移动，证明混合delta等价的判定条件，分析其与经典链环不变量的关系，还生成相关链环图并定义计算了两类delta分裂数。

Classification of $n$-component Brunnian links up to $C_n$-move

We give a classification of $n$-component links up to $C_n$-move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of trivial link and Milnor's links.

On generating sets of Yoshikawa moves for marked graph diagrams of surface-links

A marked graph diagram is a link diagram possibly with marked $4$-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two marked graph diagrams representing equivalent surface-links are related by a finite sequence of these Yoshikawa moves. In this paper, we provide some generating sets of Yoshikawa moves on marked graph diagrams representing unoriented surface-links, and also oriented surface-link…

Local moves for links with common sublinks

A C_k-move is a local move that involves (k+1) strands of a link. A C_k-move is called a C_k^d-move if these (k+1) strands belong to mutually distinct components of a link. Since a C_k^d-move preserves all k-component sublinks of a link, we consider the converse implication: are two links with common k-component sublinks related by a sequence of C_k^d-moves? We show that the answer is yes under certain assumptions, and provide explicit counter-examples for more general situations. In particular, we consider (n,k)-Brunnian links, i.e. n-component links whose k-component sublinks are all trivial…

虚拟纽结与链环的虚拟化 Δ 移动

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。

The Dabkowski-Sahi invariant and $4$-moves for links

Dabkowski and Sahi defined an invariant of a link in the $3$-sphere, which is preserved under $4$-moves. This invariant is a quotient of the fundamental group of the complement of the link. It is generally difficult to distinguish the Dabkowski-Sahi invariants of given links. In this paper, we give a necessary condition for the existence of an isomorphism between the Dabkowski-Sahi invariant of a link and that of the corresponding trivial link. Using this condition, we provide a practical obstruction to a link to be trivial up to $4$-moves.

Natural Moves for Knots and Links

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.

Writhe polynomials and shell moves for virtual knots and links

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.

Exchange moves and braid representations of links

We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one admitting an exchange move.

$Ξ$-移动下$2$-分支虚链环的分类

Classification of $2$-component virtual links up to $Ξ$-moves

本文通过改进奇缠绕数和链环数，将Taniguchi等人关于$Ξ$-移动刻画虚纽结奇缠绕数的结果扩展，完成了$Ξ$-移动下$2$-分支虚链环的分类。

On some moves on links and the Hopf crossing number

We consider arrow diagrams of links in $S^3$ and define $k$-moves on such diagrams, for any $k\in\mathbb N$. We study the equivalence classes of links in $S^3$ up to $k$-moves. For $k=2$, we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a $k$-th primitive root of unity is unchanged by a $k$-move, when $k$ is odd. It is multiplied by $-1$, when $k$ is even. It follows that, for any $k\ge 5$, there are infinitely many classes of knots modulo $k$-moves. We use these results to study the Hopf crossing number. In particular, we show t…

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。

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。

t_k-moves on links

It is a natural question to ask whether two links are equivalent by the following moves -- parallel parts of a link are changed to k-times half-twisted parts and if they are, how many moves are needed to go from one link to the other. In particular if k=2 and the second link is a trivial link it is the question about the unknotting number. The new polynomial invariants of links often allow us to answer the above questions. Also the first homology groups of cyclic branch covers over links provide some interesting information. In the first part of the paper we apply the Jones-Conway (Homflypt) a…

Triple-crossing number and moves on triple-crossing link diagrams

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n>1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-c…

定向虚拟纽结与链环的虚拟化 Delta、sharp 和 pass 移动

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。

链环在扣环通过移动下的分类

The classification of links up to clasp-pass moves

本文利用夹子理论完整分类了链环在扣环通过移动下的等价类，并将其推广到带通过与带#移动，统一并推广了多个局部移动的部分结果。

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…

Movie moves for singular link cobordisms in 4-dimensional space

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

关于链环图关于Reidemeister move I和III的有序序列

On ordered sequences for link diagrams with respect to Reidemeister moves I and III

本文证明了存在无数对无法通过特定I和III序列变换的平凡纽结图，并提出I-广义有序序列可连通所有经有限次I和III变换的链环图。

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。

纽结与链环的一二路通过移动

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。

Generalized virtualization on welded links

Let $n$ be a positive integer. The aim of this paper is to study two local moves $V(n)$ and $V^{n}$ on welded links, which are generalizations of the crossing virtualization. We show that the $V(n)$-move is an unknotting operation on welded knots for any $n$, and give a classification of welded links up to $V(n)$-moves. On the other hand, we give a necessary condition for which two welded links are equivalent up to $V^{n}$-moves. This leads to show that the $V^{n}$-move is not an unknotting operation on welded knots except for $n=1$. We also discuss relations among $V^{n}$-moves, associated co…

通过plat表示研究链环：分裂链环与复合链环

Studying links via plats: split and composite links

本文研究如何将分裂或复合链环的任意plat表示转换为可明显识别的表示，证明口袋移动是分裂链环表示的唯一障碍，并利用翻转移动对复合链环得出类似结论。

基于双quandle的浸入曲面链不变量、吉川定向第五变换与带边2-结

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。

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。

Local-move-identities for the Z[t,t^{-1}]-Alexander polynomials of 2-links, the alinking number, and high dimensional analogues

A well-known identity (Alex+) - (Alex-)=(t^{1/2}-t^{-1/2}) (Alex0) holds for three 1-links L+, L-, and L0 which satisfy a famous local-move-relation. We prove a new local-move-identity for the Z[t,t^{-1}]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander-Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the Z[t,t^{-1}]-Alexander polynomials of 2-links and the alinking number of 2-links. We show high dimensional analogu…

Independence of Roseman moves including triple points

Roseman moves are seven types of local modification for surface-link diagrams in $3$-space which generate ambient isotopies of surface-links in $4$-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of any surface-link, we construct a new diagram of the same surface-link such that any sequence of Roseman moves between them must contain moves involving triple points (and the numbers of triple points of the two diagrams are the same). Moreover, we can find a pair of two diagrams of…