High distance knots
Yair Minsky, Yoav Moriah, Saul Schleimer
我们在任意亏格中用任意高距离的Heegaard分裂构造S^3中的纽结。作为应用，对于任意正整数t和b，我们在三球面中找到了一个隧道数t的纽结，它没有（t，b）-分解.
We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.
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On hyperbolic knots realizing the maximal distance between toroidal surgeries
对于3-球面中的双曲结，环形手术之间的距离最多为5，8字形结除外。在本文中，我们确定所有双曲结，承认两个环形手术的距离为5。
We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$.
Gordian distance and Vassiliev invariants
两个结之间的戈尔迪距离测量了将一个结转换为另一个结需要多少交叉变化。众所周知，在一对戈尔迪距离为2的纽结之间总是有无穷多个不等价的纽结。在本文中，我们证明了这一事实的一个极端的概括：有任意规定的Vassiliev不变量之间的每对结的戈尔迪距离2结。
We give an obstruction for genus one knots $K$, $K'$ to have the Gordian distance one by using the $0$th coefficient of the HOMFLT polynomials. As an application, we give a new constraint for genus one knot to admit a (generalized) cosmetic crossing. Combining known results, we prove the (generalized) cosmetic crossing conjecture for genus one pretzel knots.
We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant.
Surface distance on knots
这篇论文已被作者撤回。作者发现这里的主要结果已由K. Taniyama和A. Yasuhara `On $C$ -节的距离。Kobe J. Math.11（1994），no.1，117- 127. MR1309997（95 j：57010）他感谢M。小泽的信息。
Cobordism distance on the projective space of the knot concordance group
本文定义了纽结协边群的射影空间及其整数值度量，通过环面纽结探索其基本性质，并利用扭结证明相关单纯复形包含任意大维度的无限单纯形集合。
Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra but are not high distance, by computing the bridge spectra of certain pretzel knots and Montesinos knots.
Distance between toroidal surgeries on hyperbolic knots in the 3-sphere
对于3-球面中的双曲纽结，最多1000次Dehn手术产生非双曲3-流形。作为这种特殊手术的典型情况，环面手术是产生包含不可压缩环面的闭合3-流形的手术。对应于环形手术的斜率，称为环形斜率，已知为积分或半积分。我们表明，除了8字形结的双曲结的两个积分环形斜率之间的距离是最多4。因此，任何双曲结最多允许5次环形手术。
An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, the alternation number and the alternating genus, are not comparable.
Multiple bridge surfaces restrict knot distance
设M是闭不可约可定向三维流形，K是M中的纽结，P和Q是K的桥曲面，且K关于Q不可去。证明了Q与P或$d(K,P) \leq 2-χ(Q-K)$等价.如果K不是二桥纽结，那么即使K相对于Q是可去的，结果也成立。作为一个推论，我们表明，如果一个结在3-球面有高的距离，相对于一些桥球和低的桥数，那么结有一个唯一的最小桥的位置。
Distance of Heegaard splittings of knot complements
设K是闭可定向不可约3-流形M中的纽结，P是亏格至少为2的纽结补的Heegaard分裂.设Q是K的桥曲面。然后，要么\begin{itemize} \item $d(P)\leq 2-χ(Q-K)$, or \item K can be isotoped to be disjoint from Q so that after the isotopy Q is a Heegaard surface for the knot exterior and is isotopic to a possibly stabilized copy of P. \end{itemize}
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。
The cobordism distance between knots, d(K,J), equals the four-genus g_4(K # -J). We consider d(K,K^r), where K^r is the reverse of K. It is elementary that 0 \le d(K,K^r) \le 2g_4(K) and it is known that there are knots K for which d(K,K^r) is arbitrarily large. Here it is shown that for any knot for which g_4(K) = g_3(K) (such as non-slice knots with g_3(K) = 1 or strongly quasi-positive knots), one has that d(K,K^r) is strictly less that twice g_4(K). It is shown that for arbitrary positive g, there exist knots for which d(K,K^r) = g = g_4(K). There are no known examples for which d(K,K^r) >…
We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of the manifold-knot pair. We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of a knot.
We show that for any given closed orientable 3-manifold M with a Heegaard surface of genus g, any positive integers b and n, there exists a knot K in M which admits a (g,b)-bridge splitting of distance greater than n with respect to the Heegaard surface except for (g,b) = (0,1), (0,2).
Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminusη(K)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to the…
We prove that if the lens space $L(n, 1)$ is obtained by a surgery along a knot in the lens space $L(3,1)$ that is distance one from the meridional slope, then $n$ is in $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. This result yields a classification of the coherent and non-coherent band surgeries from the trefoil to $T(2, n)$ torus knots and links. The main result is proved by studying the behavior of the Heegaard Floer $d$-invariants under integral surgery along knots in $L(3,1)$. The classification of band surgeries between the trefoil and torus knots and links is motivated by local reconnection proces…
If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an upper bound for the bridge number of K with respect to T. In particular, a nontrivial, aspherical, and atoroidal knot K with such a bridge surface has its bridge number bounded by 5 if K has a non-trivial reducing surgery; 6 if K has a non-trivial toroidal surgery; and 4g + 2 if K is null-homologous and has Seifert genus g.
Distance and bridge position
J. Hempel对Heegaard曲面距离的定义推广到了一个纽结的复杂性，该纽结相对于Heegaard曲面处于桥接位置。我们的主要结果是，在桥位置的纽结的距离是有界的两倍的属，加上边界组件的数量，在结补的本质表面。因此，通过足够高的功率的伪Anosov映射构建的结有最小的桥梁介绍是薄的。
We call a knot $K$ a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from $K$. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases. Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo's unknotting number data on 11 and 12 crossing knots. Licko…
Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal), (lens, Seifert fibred spaces) pairs. In light of Baker's work, the classification in this paper conjecturally accounts for 'most' hyperbolic knots in S^3 realizing the maximal distance between these exceptional pairs. All examples obtained in our clas…
Estimating distances in simplicial complexes with applications to 3-manifolds and handlebody-knots
本文研究了低维流形相关单纯复形中的距离关系，获得了基于分量数和曲线复形距离的距离界，并利用Heegaard分解的分裂距离为闭3-流形和柄体结定义了在稳定化下收敛于非平凡极限的新不变量。
Errata: A note on orientation-reversing distance one surgeries on non-null-homologous knots
本文是对非零调纽结上反向距离一手术相关论文的勘误，指出并报告了原文献中的一个错误。
The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the l1-norm. We show analogous results to those of Gromov, Pardon and Blair-Campisi-Taylor-Tomova about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and that there are minimal lattice-stick number knot conformations with arbitrarily high distortion.
We show that the distance of a link $K$ with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries. We further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of Seifert genus, or indeed by the genus of (almost) any essential surface or Heegaard surface in the surgered…
A three-variable transcendental invariant of planar knotoids via Gauss diagrams
该研究通过高斯图索引函数提出平面类纽结的三变量超越不变量，证明其为一阶Vassiliev不变量，并用于给出同伦平面类纽结Gordian距离的下界。
We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the X-action on Lee's perturbation of Khovanov homology.
We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.
Inscribing smooth knots with regular polygons
一个规则的$n$边形内接一个纽结是一个纽结上的一系列$n$点，使得相邻点之间的距离都相同。证明了对于任意$n$，任意光滑纽结都内接于正则$n$ -边形.
Exceptional surgery on knots
设$M$是一个不可约的、紧的、连通的、可定向的三维流形，其边界是一个环面.我们表明，如果$M$是双曲的，那么它承认最多6个有限/循环填充的最大距离5。此外，有限/循环填充到循环填充的距离至多为2。如果$M$有一个非边界平行的，不可压缩的环面，并且不是一个广义的1-迭代环面纽结补，那么最多有三个最大距离为1的有限/循环填充。此外，如果$M$有一个非边界平行的，不可压缩的环面，并且不是一个广义的1-或2-迭代环面结补，如果$M$允许奇数阶的循环填充，则$M$不允许任何其他有限/循环填充。有限/循环填充和其他例外填充之间的关系进行了讨论。
Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus greater than or equal to 2, then the coarse mapping class group of (S,V,W) is iso…
Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer $m$ we find a pair of 2-knots in the 4-sphere whose stabilization distance equals $m$. Next, using a generalized stabilization distance that counts connected sum with arbitrary 2-knots as distance zero, for every nonnegative integer $m$ we exhibit a knot $J_m$ in the 3-sphere with two slice discs in the 4-ball whose generalized stabilization d…
We extend the tangle model, originally developed by Ernst and Sumners, to include composite knots. We show that, for any prime tangle, there are no rational tangle attachments of distance greater than one that first yield a 4-plat and then a connected sum of 4-plats. This is done by building on results on exceptional Dehn fillings at maximal distance. We then apply our results to the action of the Hin recombinase on mutated sites. In particular, after solving the tangle equations for processive recombination, we use our work to give a complete set of solutions to the tangle equations modelling…
We show that on a hyperbolic knot $K$ in $S^3$, the distance between any two finite surgery slopes is at most two and consequently there are at most three nontrivial finite surgeries. Moreover in case that $K$ admits three nontrivial finite surgeries, $K$ must be the pretzel knot $P(-2,3,7)$. In case that $K$ admits two noncyclic finite surgeries or two finite surgeries at distance two, the two surgery slopes must be one of ten or seventeen specific pairs respectively. For $D$-type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulti…
Surgical distance between lens spaces
众所周知，任何一对闭可定向三维流形都可以通过一个关于纽结的Dehn手术的有限序列联系起来。此外，川内表明，这种结可以采取双曲线。本文研究了连通一对三维流形，特别是连通一对透镜空间的序列的最小长度。
Stuck Knots: Rigidity, Invariants, and Unsticking Distance
本文研究了受刚性约束的结，提出刚性不变量和解开距离，揭示了刚性约束在结简化中的作用。
Generating Infinitely Many Hyperbolic Knots with Plats
本文借助 plat 位置、辫群动力学与二重分支覆盖的 Heegaard 距离，利用 pseudo-Anosov 辫构造无穷多互不相同的素双曲 n-桥纽结，并给出亏格与体积的线性下界。
Stabilization distance bounds from link Floer homology
本文利用Heegaard Floer同调为4球中边界固定的曲面对构造新不变量，给出稳定化距离和双重点距离的下界，并借此找到距离任意大的切片圆盘对及非0-协边切片圆盘。
Arc distance equals level number
在一个3-流形中，一个相对于亏格g Heegaard曲面处于1-桥位置的纽结K可以通过合痕移动通过处于1-桥位置的纽结，直到它位于由n-1个直管连接在一起的n个平行亏格g曲面的并集中，K与每个直管相交于连接两端的两个弧中。我们证明，这是可能的最小n等于一个Hempel型距离不变定义使用弧复杂的两个洞genus-g表面
We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has $n - 1$ rows of twists and all its exponents have absolute value greater than or equal to three then the distance of the bridge sphere is exactly $\lceil n/(2(m - 2)) \rceil$, where $\lceil x \rceil$ is the smallest integer greater than or equal to $x$. As a corollary, we conclude that if such a diagram has more than $4m(m-2)$ rows then the bridge sphere defining the plat projection is the unique mi…
We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve in $\mathbb{C}^2$ with the unit 4-ball from which a 4-ball of smaller radius is removed. Connections to the realization problem of $A_n$-singularities on algebraic plane curves and the adjacency problem for plane curve singularities are discussed. To obstruct the existence of cobordisms, we use Ozsváth, Stipsicz, and Szabó's $Υ$-invariant, which we provide …
A Class of Curves In Every Knot Type Where Chords of High Distortion are Common
曲线的变形是指曲线上不同点对上弧长与点间空间距离之比的上确界。格罗莫夫在1981年提出了一个问题，是否每一种纽结类型的曲线都可以构造成失真小于一个普适常数C。要解决格罗莫夫问题，似乎需要根据某些拓扑不变量构造纽结变形的下界。我们试图使这样的界限更容易构造表明，对高失真的点是非常常见的曲线上的最小长度的曲线在一个给定的结类型的曲线上有界的扭曲和扭曲厚度的上限以下。
On the distance between Seifert surfaces
对于纽结$K$，Kakimizu引入了一个单纯复形，其顶点都是$K$的极小亏格生成曲面的合痕类.本文的第一个目的是证明当$K$是超环面时，该复形的1-骨架的直径在纽结亏格中为二次函数。证明了当$K$为超环面时，$K$的两个极小亏格生成曲面的相交数在纽结亏格中也是有界的.作为一个应用，我们证明了Kakimizu复形在所有环面亏格1结点之间的简单连通性。
Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $ K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$. Though $t(K^{\star})\leq t(K)+1$, Example 1.1 in Section 1 shows that in some case, $ t(K^{\star})\leq t(K)$. So it is interesting to know when $t(K^{\star})= t(K)+1$. After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot $K^{\star}$ and its companion $K$, $t(K^{\star})\geq t(K)$; (2) if either $K$ admits a high distance H…
An Upsilon torsion function for knot Floer homology
本文提出一个参数化的Upsilon扭矩函数，用于研究结之间的cobordism性质及Gordian距离的障碍条件。
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。
An exploration of low crossing and chiral cosmetic bands with grid diagrams
本文通过网格图方法研究低交叉数结间的非相干带连接，发现多个新的H(2)-距离为一的配对，并解释了手性cosmetic带的构造。
High distance Heegaard splittings via fat train tracks
我们定义了“胖”训练轨道，并利用它们给出了封闭可定向三维流形Heegaard分裂的Hempel距离的一个组合判据。我们应用这个标准3-流形从手术结在三个球。
This is an introductory article on high dimensional knots for the beginners. High dimensional knot theory is an exciting field. It is a field of knot theory, which is one of topology and is connected with many ones. In this article we use few literal expressions, equations, functions, etc. We barely suppose that the readers have studied manifolds, homology theory, or topics beyond them. Is there a nontrivial high dimensional knot? We first answer this question. After that, we explain local moves on high dimensional knots and the projections of high dimensional knots.
We show that if $K$ is a knot in $S^3$ and $Σ$ is a bridge sphere for $K$ with high distance and $2n$ punctures, the number of perturbations of $K$ required to interchange the two balls bounded by $Σ$ via an isotopy is $n$. We also construct a knot with two different bridge spheres with $2n$ and $2n-1$ bridges respectively for which any common perturbation has at least $3n-1$ bridges. We generalize both of these results to bridge surfaces for knots in any 3-manifold.
Bridge Number and the Curve Complex
我们发现，有双曲线隧道号码一个结任意高的桥数和“大多数”隧道号码一个结是不是一个桥相对于一个unknotted环面。证明依赖于亏格为2的曲面的曲线复形中桥数与一定距离之间的关系。
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 construct a map from knots to (abstract) 2-knots which can be extended to higher dimensions; this map is the natural "knot" counterpart for "braid" theory of groups $G_{n}^{k}$.
We give necessary and sufficient conditions for an integer to be the signature of a (4q-1)-knot in the (4q+1)-sphere with a given square-free Alexander polynomial.
Smoothly slice knots with Alexander polynomial 1 and high unknotting number
本文证明存在一个同时具有光滑双切片性、双向手性、Alexander多项式为1且解结数为5的纽结，揭示了这些拓扑性质可以共存。
We show the existence of infinitely many prime knots each of which having in their complements meridional essential surfaces with two boundary components and arbitrarily high genus.
Given a link in $S^3$ we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restricting the knot types which can arise from a sequence of splitting operations on a link. This allows us to answer a question asked by Colin Adams in 1996.
We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.
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