Haruko Aida Miyazawa, Akira Yasuhara
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.
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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 construct two families of satellite constructions for Brunnian links, called the satellite sum and the satellite tie. An interesting fact is that by applying the satellite sum and the satellite tie constructions, we can build infinitely many new Brunnian links from any given Brunnian links. With the helps of the satellite sum and the satellite tie, we give a new decomposition theorem for Brunnian links. We prove that every Brunnian link determines a unique labelled "tree-arrow structure" such that each vertex of the tree represents a generalized Hopf link, a hyperbolic Brunni…
Hyperbolicity of Brunnian links
本文提出基于几何/组合拓扑的新技术，用于判定至少含一个未打结分量的链环的双曲性，并据此确定了文献中所有Brunnian链环的双曲性，发现了许多无限族双曲Brunnian链环。
We show some properties of a Seifert matrix of an $n$-component Brunnian link. In particular, we give a necessary and sufficient condition for a matrix to be a Seifert matrix of a 2-component Brunnian link up to S-equivalence.
In this article, it is proved that the eigenvalue variety of the exterior of a nontrivial, non-Hopf, Brunnian link in $\mathbb{S}^{3}$ contains a nontrivial component of maximal dimension. This generalises, for Brunnian links, the nontriviality of the $A$-polynomial of a nontrivial knot in $\mathbb{S}^{3}$.
Brunnian links have been known for a long time in knot theory, whereas the idea of n-triviality is a recent innovation. We illustrate the relationship between the two concepts with four short theorems.
We consider surgery moves along (n+1)-component Brunnian links in compact connected oriented 3-manifolds, where the framing of the each component is 1/k for k in Z. We show that no finite type invariant of degree < 2n-2 can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We relate finite type invariants of integral homology spheres obtained by such operations to Goussarov-Vassiliev invariants of Brunnian links.
This paper proves that convex Brunnian links exist for every dimension $n \geq 3$ by constructing explicit examples. These examples are three-component links which are higher-dimensional generalizations of the Borromean rings.
A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, the first author proved that the restriction to Brunnian links of any Goussarov-Vassiliev finite type invariant of (n+1)-component links of degree<2n is trivial. The purpose of this paper is to study the first nontrivial case. We will show that the restriction of an invariant of degree 2n to (n+1)-component Brunnian links can be expressed as a quadratic form on the Milnor link-homotopy invariants of length n+1.
Let $M_λ$ be the $λ$-component Milnor link. For $λ\ge 3$, we determine completely when a finite slope surgery along $M_λ$ yields a lens space including $S^3$ and $S^1\times S^2$, where {\it finite slope surgery} implies that a surgery coefficient of every component is not $\infty$. For $λ=3$ (i.e.\ the Borromean rings), there are three infinite sequences of finite slope surgeries yielding lens spaces. For $λ\ge 4$, any finite slope surgery does not yield a lens space. As a corollary, $M_λ$ for $λ\ge 3$ does not yield both $S^3$ and $S^1\times S^2$ by any finite slope surgery. We generalize the…
We prove that if n\ge1, then an (n+1)-component Brunnian link L in a connected, oriented 3-manifold is C_n-equivalent to an unlink. We also prove that if n\ge2, then L can not be distinguished from an unlink by any Goussarov-Vassiliev finite type invariant of degree<2n.
The purpose of the paper is twofold. First, we give a short proof using the Kontsevich integral for the fact that the restriction of an invariant of degree 2n to (n+1)-component Brunnian links can be expressed as a quadratic form on the Milnor mu-bar link-homotopy invariants of length n+1. Second, we describe the structure of the Brunnian part of the degree 2n-graded quotient of the Goussarov--Vassiliev filtration for (n+1)-component links.
Brunnian links of 3-balls in the 4-sphere
本文在4维球面中构造了无限多个n个组成部分的3球体布鲁恩链，并提供了相关分割球存在的新证明。
If L_1 and L_2 are two Brunnian links with all pairwise linking numbers 0, then we show that L_1 and L_2 are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If L_1 is a Brunnian link with all pairwise linking numbers 0, and the complement of L_2 is homeomorphic to the complement of L_1, then we show that L_2 may be obtained from L_1 by a sequence of twists around unknotted components. Finally, we show that for any positive integer n, an algorithm for detecting an n-component unlink leads immediately …
Adjacency of three-manifolds and Brunnian links
本文研究了三维流形之间的邻接性概念，通过链环的手术操作定义了流形间的邻接关系，并类比结的邻接性结果进行特征化。
Brunnian exotic surface links in the 4-ball
本文在4-球中构造了两种Brunnian exotic曲面链环对，并推广了卫星操作与覆盖链环工具，以调控分支数、亏格及解链所需移除的分支数。
We present two practical and widely applicable methods, including some criteria and a general procedure, for detecting Brunnian property of a link, if each component is known to be unknot. The methods are based on observation and handwork. They are used successfully for all Brunnian links known so far. Typical examples and extensive experiments illustrate their efficiency. As an application, infinite families of Brunnian links are created, and we establish a general way to construct new ones in bulk
Habiro gave principal ideals of Z[q,q^{-1}] in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links and Brunnian links are contained in smaller ideals of Z[q,q^{-1}] generated by several elements. In this paper, we prove that these ideals also are principal, each generated by a product of cyclotomic polynomials.
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.
A link L is called Brunnian if every proper sublink of L is trivial. Similarly, a bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we give a small subalgebra of the n-fold completed tensor power of U_h(sl_2) in which the universal sl_2 invariant of n-component Brunnian bottom tangles takes values. As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.
We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres…
On a family of hyperbolic Brunnian links and their volumes
本文构造推广 Debrunner 例子的一族多分支 Brunnian 链环，并利用带尖点流形的 Dehn 填充给出其补空间双曲体积上界。
We provide an alternative proof that Koschorke's $κ$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $μ$--…
Brunnian links and Kontsevich graph complex I
本文通过构造链映射，将Kontsevich图复形与流形的链复形联系起来，并提出新的元素构造方法。
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…
We extend the well-known Borromean and Brunnian rings to new higher order versions. Then we suggest an extension of the connection between Efimov states in cold gases and Borromean and Brunnian rings to these new higher order links. This gives rise to a whole new hierarchy of possible states with Efimov states at the bottom.
We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length < k are all zero into a link with vanishing Milnor invariants of length < 2k, and we provide formulas for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Delta-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if a link L with…
A recipe for exotic 2-links in closed 4-manifolds whose components are topological unknots
本文提出了一种在闭单连通4-流形中构造无限多组拓扑同痕、光滑不等价且分支拓扑平凡的2-链环的方法，提供了文献中首批此类例子，并揭示了链环在产生奇异现象中的关键作用。
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.
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…
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 …
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.
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。
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…
We classify the Montesinos links up to mutation and 5-move equivalence, and obtain from this a Jones and Kauffman polynomial test for a Montesinos link.
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…
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…
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。
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.
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.
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 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.
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。
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…
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。
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。
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…
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…
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。
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。
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…
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.
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。
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。
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。
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。
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…
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。
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。
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。
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