Differently knotted symplectic surfaces in D^4 bounded by the same transverse knot
Andrew Geng
本文证明了在4-球中存在两个辛曲面，这两个辛曲面与同一个横结有界，具有相同的拓扑结构（如抽象曲面），并通过它们的补曲面的基本群来区分。
In this paper we show that there are two symplectic surfaces in the 4-ball which bound the same transverse knot, have the same topology (as abstract surfaces), and are distinguished by the fundamental groups of their complements.
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Topological triviality of smoothly knotted surfaces in 4-manifolds
已知Fintushel-Stern边缘手术的一些推广和变体可产生光滑的打结表面。我们证明了如果它们的补群的基本群是循环的，那么这些曲面是拓扑无结的。使用一个扭曲旋转建设从高维纽结理论，我们构造的例子，纽结表面的补充循环基本群。
Lagrangian Surfaces in a Fixed Homology Class: Existence of Knotted Lagrangian Tori
在本文中，我们证明了存在单连通辛4-流形，它包含无穷多个纽结拉格朗日环面，即环面的拉格朗日嵌入是同伦的，但不是同位素的。此外，它们所表示的同调类可以被假定为非平凡的和本原的。这回答了Eliashberg和Polterovich的一个问题。
Parametrizing some interesting knotted surfaces
本文提出了一种参数化结表面的方法，并给出了具体示例。
Knotted solid tori in contact manifolds
该研究聚焦切触流形中的实心环面，探究纽结型宽度与最大Thurston-Bennequin不变量的关系并给出判定条件，还证明多类纽结型中存在大量此前仅在$S^3$中发现的不可加厚环面。
Twist Spinning Knotted Trivalent Graphs
1965年，E. C.塞曼证明了（n-1）-空间中任意纽结球的（+/-）-扭转自旋在n-球中是非纽结的。1991年，Y. Marumoto和Y. Nakanishi用运动图像法给出了塞曼定理的另一种证明。本文定义了一个二维纽结泡沫，它是纽结球的推广，并证明了一个纽结三价图的（+/-）-扭自旋可以是纽结的.然后，我们构造了一些家庭的打结图的（+/-）-扭转自旋总是unknotted。
Polynomially knotted 2-spheres
本文回顾经典纽结的多项式参数化，证明长2维纽结的类似结果，并为旋转与扭转旋转等纽结球面构造多项式参数化。
Knotted handle decomposing spheres for handlebody-knots
我们证明了一个外部是边界不可约的实体结有唯一的最大非嵌套集的纽结手柄分解领域的同位素和环移动。作为一个应用，我们证明了体结$6_{14}$和$6_{15}$是不等价的.我们还证明了某些亏格的两个带纽结柄的纽结分解球可以由它们的外部确定。作为一个应用，我们证明了$6_{14}$和$6_{15}$的外部不是同胚的.
The Meridional Rank Conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the analogous conjecture for knotted surfaces in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order and which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, …
Stably irreducible non-orientable knotted surfaces
本文提出了四维球面中纽结曲面可约性的基本障碍，并据此构造了稳定不可约的不可定向曲面。
This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a new construction of exotically knotted surfaces, including exotic slice surfaces of all genera in the 4-ball and exotic closed surfaces in larger 4-manifolds. The construction is well-suited to the complex and symplectic settings, providing the first examples of exotically knotted complex curves and symplectic 2-spheres. Along the way, we articulate some di…
A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are the only intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattman and Oh showed that there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Fur…
New lower bound for the distortion of a knotted curve
证明了$\R^3$中纽结曲线的畸变大于4.76。这改进了John M. Sullivan和Elizabeth Denne在\cite{DS}。
A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are the only intrinsically knotted graphs with 21 edges. In this paper we present the following results: there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Two are the cousins 94 and 110 …
In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. We show that for every handlebody-knot pair associated to a genus two "prime bi-knotted" surface, one is irreducible, and the other is reducible. Furthermore, for given two genus two handlebody-knots $H_1$ and $H_2$ satisfying certain conditions, we will construct a "prime bi-knot…
Computations of Quandle Cocycle Invariants of Knotted Curves and Surfaces
Laurel Langford和作者在Math.GT/9903135中利用Quandle上同调引入了纽结曲线曲面的状态和不变量，本文给出了计算这类不变量的方法和实例计算。本文给出了一些量子点的上同调维数的计算机计算。对于经典的纽结，Burau表示与Maple程序一起被用来评估纽结表的不变量。对于四维空间中的纽结曲面，使用了电影方法和曲面辫理论。讨论了纽结的不变量与对称性之间的关系。
Bridge trisections of knotted surfaces in $S^4$
我们在四球面中引入了纽结曲面的桥三分性。这种描述的灵感来自于盖伊和柯比关于四流形的三分性的工作，并将三个球面中链接的桥分裂的经典概念扩展到四维。我们证明，每一个打结的表面在四球承认一个桥三分（分解成三个简单的部分），任何两个桥三分为一个固定的表面相关的序列的稳定和不稳定。我们还介绍了相应的图示表示打结的表面，并描述了一组移动，足以通过两个图之间的同一表面。使用这些分解，我们定义了一个新的复杂性措施：一个纽结表面的桥数。此外，我们还对低复杂度的桥三分性进行了分类，并将桥三分性与纽结曲面补的基本群联系起来，证明了存在桥数任意大的纽结曲面.
A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface K defines a projection of K onto a 2-sphere, whose set of critical values is the hyperbolic diagram of K. We apply such projections, called flattenings, to define three invariants of knotted surfaces: the layering, the trunk and the partition number. The basic properties of flattenings and their derived invariants are obtained. Our construction is used to s…
On symmetry and exterior problems of knotted handlebodies
研究3维球面中亏格2纽结柄体的对称性与外部问题，确定由非整数环面Dehn手术双曲纽结构造的柄体对称群并解决其外部问题，同时推广Lee-Lee族。
We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.
A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that there are exactly 14 intrinsically knotted graphs with 21 edges, in which the Heawood graph is the only bipartite graph. The authors showed that there are exactly two graphs with at most 22 edges that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110 of the $E_9+e$ family. In this paper we show that there are exactly six bipartite intrinsically knotted graphs with 23 edges so that every vertex …
Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology
在本文中，我们介绍了一种表示嵌入打结（有时拉格朗日）环面在$\BR^4$称为超立方体图，即，四维立体图我们证明了超立方体同调的存在性，这种同调在四维立方体图移动下是不变的，这种同调是基于纽结弗洛尔同调的。我们提供的例子超立方体图和超立方体的同源性，包括使用新的不变量区分（立方体移动）两个“霍普夫链接”环面。我们还给出了“三叶”环面和浸没打结环面的例子，它是$5_2$结和三叶结的融合。
The Kinoshita graph is a particular embedding in the 3-sphere of a graph with three edges, two vertices and no loops. It has the remarkable property that although the removal of any edge results in an unknotted loop, the Kinoshita graph is itself knotted. We use two classical theorems from knot theory to give two particularly simple proofs that the Kinoshita graph is knotted. Apart from appealing to the two classical theorems, the exposition is elementary and is aimed at those with only a passing familiarity with knot theory.
Bounds for Kirby-Thompson invariants of knotted surfaces
本文研究结表面的Kirby-Thompson不变量，通过两种不同方法计算其精确值，并给出下界。
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。
Minimal Flat Knotted Ribbons
本文给出了平面结带的数学模型，并对打三叶结和8字结所需的最小丝带长度（给定宽度）作了具体的计算。第一个猜想指出（对于宽度1），系一个开放式三叶结所需的最小丝带长度为L = 4（F + 1）/Sqrt[2 + F]，其中F =（1 + Sqrt[5]）/2是黄金分割率。第二个猜想指出，8字结的最小长度（在相同的意义上）是32/Sqrt[15]。
On a banded link presentation of knotted surfaces
我们将讨论一种方法，通过检查其莫尔斯临界点的数量和位置，在四维空间中的纽结曲面的视觉表示。利用这种方法，我们将研究具有一个指数为1的临界点的曲面纽结。然后，我们展示了无穷多个相互不同的曲面节点，它们具有两个指数为1的临界点的嵌入。接下来，我们定义一个长平面的带状链接的任何表面结，并显示图解长平面形式的$n$ -扭转旋转$(2,t)$ -环面结。
We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move. We show that ternary quasigroups satisfying these axioms appear naturally in invariants of Reidemeister, Yoshikawa, and Roseman moves. Our homology has a degenerate subcomplex. The normalized homology yields invariants of knots and knotted surfaces.
Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use link homology to construct an invariant of knotted surfaces (up to isotopy) which distinguishes the unknotted sphere from certain knotted spheres. We also construct an invariant of a bridge-trisected surface in the the form of an $A_\infty$-algebra. Both invariants are defined by a novel connection between $A_\infty$-algebras and Manolescu and Ozsváth's hyp…
Knotted Hamiltonian cycles in linear embedding of $K_7$ into $\mathbb{R}^3$
1983年Conway和Gordon证明了完全图$K_7$到$\mathbb{R}^3$的任何嵌入都至少包含一个非平凡纽结作为它的哈密顿圈。在他们的工作结（也链接）被认为是抽象图的内在属性，许多后续的工作一直持续到最近。在本文中，我们对$K_7$的线性嵌入中的打结哈密顿圈感兴趣。具体地说，它表明，$K_7$的任何线性嵌入包含最多三个数字8结作为其哈密顿圈。
Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An important problem in knot theory is to study the minimal ratio of length to width of a flat knotted ribbon. This minimal ratio is called the ribbonlength of the knot. It has been conjectured that the ribbonlength has an upper bound and a lower bound which are both linear in the crossing number of the knot. In the first part of the paper, we use…
For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon $2$-knot is not even symmetric in general. Via an alternative notion of ribbon $2$-knots, we give a topological condition on a $2$-knot that implies the factorization of the Alexander polynomial.
On unique decomposition of knotted handlebodies
本文研究了3-球中打结柄体沿分解2-球因式分解的唯一性问题，在特定条件下证明了唯一性，并据此确定了$6_{10}$的手性，构造了具有同胚外部的双曲柄体结无穷族。
Alexander Numbering of Knotted Surface Diagrams
给出了三重点、分支点及其与无穷远距离的关系式。我们恢复平凡正常的欧拉类定向曲面，和公式上签署的三重点。
Categorical Groups, Knots and Knotted Surfaces
我们从任意有限范畴群中定义了一个纽结不变量和一个2-纽结不变量。我们计算了一个显式的例子纺三叶草。
Knotted Spheres and Graphs in Balls
给定一个适当的嵌入图伽玛在一个球B和穿刺球西格玛适当地嵌入在B -伽玛，我们检查伽玛的条件是必要的，以确保西格玛是边界平行。
Bikei Invariants and Gauss Diagrams for Virtual Knotted Surfaces
标记顶点图提供了一种组合方式来表示$\mathbb{R}^4$中的打结表面;包括虚拟交叉允许虚拟打结表面和虚拟配边理论。Biquandle计数不变量的定义只为标记的顶点图表示打结可定向曲面，我们扩展这些不变量的所有虚拟标记顶点图考虑着色对合biquandles，也被称为bikei。我们引入了一种用高斯图表示标记顶点图的方法，并用它来刻画可定向性。
Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces
2捻纺三叶是一个例子，一个球，是在四维空间打结。在这里，该示例被示出为不同于具有相反取向的相同球体。为了证明这一事实，一个状态和不变的经典纽结和纽结表面开发通过上同调理论的机架和quandles（也称为分配群胚）。Quandle是一个具有二元运算的集合--其公理模型是Reidemeister在3-空间中的打结和链接曲线的经典理论中的运动。利用quandle元对纽结曲线曲面图的着色以及quandle的上圈，定义了3-空间中纽结圆和4-空间中纽结曲面的状态和不变量.本文计算了各种量子点的上同调群，并将其应用于研究经典纽结、链环及其它链环曲面的状态和不变量。对于各种纽结和链环，包括三叶结和8字形纽结，证明了不变量的非平凡性，反之，纽结不变量用于证明各种quandle的上同调的非平凡性。
In this paper we provide a new obstruction to 0-concordance of knotted surfaces in $S^4$ in terms of Alexander ideals. We use this to prove the existence of infinitely many linearly independent 0-concordance classes and to provide the first proof that the submonoid of 2-knots is not a group. The main result is that the Alexander ideal induces a homomorphism from the 0-concordance monoid $\mathscr{C}_0$ of oriented surface knots in $S^4$ to the ideal class monoid of $\mathbb{Z}[t^{\pm1}]$. Consequently, any surface knot with nonprincipal Alexander ideal is not 0-slice and in fact, not invertibl…
Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial
这是一系列研究w-打结物体（w-结，w-辫子，w-缠结等）的论文中的第一篇，这使得一类打结的物体比它们通常的对应物更复杂但更脆弱。W-braids群由Fenn-Rimanyi-Rourke研究（称为“{w}elded braids”），并被证明同构于自由群Fn的“基共轭”自同构的McCool群。Brendle-Hatcher，追溯到Goldsmith，已经证明这组是R3中的一组飞行环的电影。Satoh研究了几类w-纽结对象（作为“{w}eakly-virtual”），并证明它们与R4中的某些纽结曲面密切相关。所以w-纽结对象在代数和拓扑上都很有趣。本文研究w-纽结对象的有限型不变量。继Berceanu-Papadima之后，我们构造了W-辫子和W-缠结的同态泛有限型不变量（“展开式”）.我们发现，普遍的有限型不变量的w-结本质上是亚历山大多项式。我们发现w-纽结对象的“箭图”空间Aw与不一定度量化的李代数有关。许多关于w-纽结对象的问题与李代数的问题是等价的。最值得注意的是，我们发现，一个同态展开的w-knotted泡沫是本质上相同的解决方案的Kashiwara-Vergne猜想（KV），从而给出了拓扑解释的工作Alekseev-Torossian工作KV和Drinfel'd缔合。不过，w结的真正价值可能会在以后出现，因为我们希望它们能作为虚拟结研究的一个例子。我们期望v-打结对象…
Smoothly knotted surfaces that remain distinct after many internal stabilizations
本文研究单连通$4$流形中光滑纽结曲面的内部稳定化，证明使曲面变得光滑同痕所需的稳定化次数无上界，提出微妙光滑纽结曲面对的定义，并证明这种现象具有普遍性。
One stabilization is not enough for closed knotted surfaces
本文证明存在包含成对异构纽结2-球面的光滑4-流形，其异构性在一次稳定化后仍保持，表明“一次即足够”定理不适用于同调类为特征类的闭曲面。
Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces
我们定义了一个不变量的焊接虚拟结从每个有限的交叉模块考虑交叉模块不变量的带状打结的表面，这是自然与他们。通过计算明确的例子，我们阐明了所获得的不变量是非平凡的。我们定义焊接虚拟图，并考虑他们以类似的方式定义的不变量。
Knotted surfaces with simply-connected complements
研究边界为固定纽结 K 且嵌入单连通四维流形的特定曲面，证明若此类曲面同调，则在相对于边界的拓扑环境同痕下是同痕的。
Knotted surfaces in 4-manifolds and stabilizations
本文研究了4-流形中奇异纽结曲面的稳定等价，这些曲面是拓扑同位素的但不是光滑同位素的。我们证明了任何一对嵌入的表面在同一同调类成为光滑的同位素稳定后，通过处理添加周围的4-流形，而且可以假设是附加在一个标准的方式（本地和unknotedly）在许多有利的情况下。特别是，任何具有环状基本群互补的奇异纽结表面对，在经过相同数量的标准稳定化之后，都变得平滑地同位素--类似于C.T.C.。Wall关于单连通4-流形的稳定等价的著名结果。此外，我们表明，所有建设的外来knottings的表面，我们知道，这显示了各种各样的技术和想法，产生表面，成为顺利同位素后，一个单一的稳定。
Knotted and linked products of recombination on $T(2,n)\#T(2,m)$ substrates
我们开发了一个拓扑模型的位点特异性重组，适用于基板是连接的形式$T(2,n)\#T(2,m)$的两个环面链接的总和。然后，我们用我们的模型来证明，所有的结和链接，可以通过位点特异性重组在这样的基板上包含在两个家庭，我们说明之一。
We present a marked analogue of Carter and Saito's movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopies, we define three equivalence relations for marked surfaces and work over an equivalence class of marked surfaces when providing our generalization of Carter and Saito's movie theorem.
Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of a protein's planar projections using topological objects called knotoids. We approach the analysis of proteins with the same topology by introducing a topologically inspired statistical metric between their knotoid distributions. We detect geometric differences between trefoil proteins by characterising their entanglement and we recover a clustering by sequence …
On the Kontsevich integral for knotted trivalent graphs
我们构建了一个扩展的Kontsevich积分的结结三价图，它与方向开关，边删除，边unzips，和连接的总和。1997年，村上春树和Ohtsuki [MO]首先构建了这样一个扩展，建立在Drinfel'd的结合子理论上。我们构造了一个逐步的定义，使用初等Kontsevich积分方法，得到一个单参数的家庭的修正，所有产生的不变量下表现良好的图形操作。
Triangulations of the 3-sphere with knotted edge
本文证明任意纽结K可嵌入单顶点3-球面三角剖分的边中，通过构造性方法基于完全增广链环生成复杂三角剖分，并验证其在常数因子内为最小规模。
Given a simply-connected closed 4-manifold $X$ and a smoothly embedded oriented surface $Σ$, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in $X$ that are pairwise homeomorphic to $Σ$, but not diffeomorphic. We prove that for all known examples of surface knots constructed from knot surgery operations that preserve the fundamental group of the complement of surface knots, they become pairwise diffeomorphic after stabilizing by connected summing with one $S^2\tilde{\times}S^2$. When $X$ is spin, we show in addition that any surfaces obtained by a knot su…
Some combinatorial aspects of movies and movie-moves in the theory of smoothly knotted surfaces in R4
对于{1，2，.}的真子集U，31}我们定义并构造了Carter-Rieger-Saito电影（表示纽结曲面）的U-正则同痕不变量，并使用这些不变量来表明存在由电影M1和M2表示的周围的同素纽结曲面，对于所述电影M1和M2，需要类型31的电影移动来从M1到达M2。
On knots that divide ribbon knotted surfaces
本文研究了半带状结的 genus 和半融合数，计算了多个结的半带状 genus 并探讨了其与双可切 genus 的关系。
We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an i…
Intrinsically Linked Graphs with Knotted Components
我们构造一个图G，使得G到R^{3}的任何嵌入都包含两个分量的非分裂链，其中至少一个分量是非平凡结。此外，对于任意m < n，我们生成一个图H，使得H的每个嵌入都包含一个非分裂的n个分量链，其中至少m个分量是非平凡的结点。然后，我们把我们的注意力转向完全图，并表明，对于任何给定的n，每一个嵌入的足够大的完全图包含两个组件链接的环绕数是一个非零倍的n。
We give an explicit construction of complex maps whose nodal line have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, $\ell$) Lissajous figure, and are therefore a subfamily of spiral knots generalising the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalising to rational maps with application to the Skyrme-F…
A zigzag in a map (a $2$-cell embedding of a connected graph in a connected closed $2$-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a common vertex, 2) for any three consecutive edges the first and the third edges are disjoint and the face containing the first and the second edges is distinct from the face which contains the second and the third. A map is $z$-knotted if it contains a single zigzag. Such maps are closely connected to Gauss code problem and have nice homological properties. We s…
In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and such that the inverse image of zero in the complex plane contains a knotted curve in three space. We show that for knots in three space this is a generic situation in that every smooth knot K in three space has a smooth classifying map f from three space to the complex plane such th…
The Second Hull of a Knotted Curve
该研究定义了闭合曲线的高阶包壳概念，证明扭结曲线的第二包壳非空，为Fary/Milnor定理给出了新证明。
A knotted minimal tree
本文在R^3中的单位3-球的边界上构造了一个有限集X，它的最小树是纽结的。这个例子回答了Rob Kirby在Michael Freedman提出的“低维拓扑问题”中的问题5.17：“给定B^3的边界上的点X的有限集，设T是B^3中包含X的最小长度的树。T是不打结的吗？也就是说，在包含T的B^3中是否存在嵌入2-球的PL？''
The Distortion of a Knotted Curve
曲线的变形测量最大弧/弦长比。Gromov证明了任何闭曲线都至少有pi/2的变形，并询问了节点的变形。在这里，我们证明了任何非平凡的驯服结的扭曲至少为5 pi/3;例子表明，7.16以下的扭曲足以建立一个三叶结。我们的论点使用的最短本质割线的存在性和边界本质弧的特性。
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