Commensurability classes containing three knot complements
Neil R. Hoffman
本文给出了一类双曲纽结补的无限族，它们在各自的可积类中有三个纽结补。
This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.
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Commensurability classes of 2-bridge knot complements
我们证明了双曲2-桥纽结补是其可分解类中唯一的纽结补。我们还讨论了双曲纽结补的构造，并提出了一个关于双曲纽结补的个数的猜想。
Commensurability of knots and L^2-invariants
我们证明了L^2-挠率和von Neumann rho-不变量导致了纽结的可扩展性不变量。
Knot commensurability and the Berge conjecture
我们研究了双曲纽结补集在一般情况下没有隐藏对称性的纽结的可扩展性类。我们表明，这种纽结补集是可循环的，并有最多$3$双曲纽结补集的循环可循环类。此外，如果两个双曲纽结有循环可交换的补数，则它们是同亏格纤维化的，并且是手征的。给出了周期纽结的补的循环可积类的一个刻画。在非周期性的情况下，我们减少了循环的可扩展性类的Berge猜想的概括的特征。
Commensurability classes of (-2,3,n) pretzel knot complements
设K是一个双曲（-2，3，n）椒盐卷饼结，M = S^3 K是它的补。对于这些纽结，我们验证了Reid和Walsh的一个猜想：在M的可积类中至多有三个纽结补.事实上，如果n \neq 7，我们表明，M是唯一的结补在其类。我们包括例子来说明我们的方法如何适用于广泛的一类Montesinos结。
Commensurability classes of twist knots
本文证明了：如果$M_K$是$\mathbb S^3$中非纤维纽结$K$的补，则在$\mathbb Z/ 2 \mathbb Z$ -同调球面中$M_K$不能成为纤维纽结补.为了证明这一结果，我们导出了扭结特征变化的递归描述，并证明了D。Calegari和N.邓菲尔德对这些品种感到满意。此外，我们部分扩展我们的结果，第二个无限家庭的2桥结。
Commensurability of 1-cusped hyperbolic 3-manifolds
我们给出的例子，非纤维双曲结互补的同源领域是不可分割的纤维结互补的同源领域。实际上，我们给出了许多同调球面上纽结补的例子，这些纽结补都具有同调球面上每个可解纽结补都具有非一阶Alexander多项式的性质。
In this paper we analyze symmetries, hidden symmetries, and commensurability classes of $(ε, d_L)$-twisted knot complements, which are the complements of knots that have a sufficiently large number of twists in each of their twist regions. These knot complements can be constructed via long Dehn fillings on fully augmented links complements. We show that such knot complements have no hidden symmetries, which implies that there are at most two other knot complements in their respective commensurability classes. Under mild additional hypotheses, we show that these knots have at most four (orienta…
We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we consider are compact, so techniques involving cusps that are typically used to distinguishing mutant pairs of knots are not applicable. Indeed, new techniques may need to be developed to study commensurability of mutant pairs of polyhedra.
Trace fields and commensurability of link complements
本文研究了作为双曲三维流形可拓不变量的迹场的强度。我们构造了一个具有相同迹场的两分量双曲链补的无限族，以及无穷多个具有相同性质的1-尖有限体积双曲三维流形。我们还证明了上述两个分量的链补有整数迹，但每个都有一个非整数迹的突变体。
Algebraic invariants, mutation, and commensurability of link complements
通过在全测地四穿孔球面上粘贴缠结，构造了一类双曲链补族，并研究了其成员间的可拓关系。那些具有不同体积的是不可区分的，通过它们的剪刀同余类来区分。突变产生任意大的非等距流形的有限子族，具有相同的体积和剪刀同余类。根据突变的选择，这些流形可能是可解的或不可解的，在后一种情况下通过尖点参数来区分。它们都有迹场Q（i，\sqrt{2}），但有些有积分迹，而有些没有。
In this paper, we show that any non-arithmetic hyperbolic $2$-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic $2$-bridge link complement cannot irregularly cover a hyperbolic $3$-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of $3$-manifolds with non-trivial JSJ-decomposition and rank two fundamental groups. We also show that the only commensurable hyperbolic $2$-bridge link complements are the figure-eight knot complement and the $6_{2}^{2}$ link complement. Our work requires a carefu…
In this paper we study some consequences of the author's classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.
Arithmeticity and commensurability of links in thickened surfaces
本文针对由常曲率曲面正则4价平铺构建的直角平铺链环，利用对称性论证等给出其算术性的完整刻画，结合典范分解与数论数据确定两两可公度条件。
Geometric limits of knot complements, II: Graphs determined by their complements
证明了三维球面中存在内部不同胚于双曲纽结补的几何极限的紧致子流形。
We consider the following question: when is the manifold obtained by gluing together two knot complements an $L$-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an $L$-space. We extend this result to allow for arbitrary integer framings. We find that splicing two integer framed nontrivial knot complements only produces an $L$-space if both knots are $L$-space knots and the framings lie in an appropriate range. The proof involves a careful analysis of the bordered Heegaard Floer invariants of each knot complement.
Totally Geodesic Seifert Surfaces in Hyperbolic Knot and Link Complements I
第一个例子完全测地塞弗特曲面构造双曲结和链接，包括自由和完全打结的表面。然后证明了两个桥纽结补不可能包含全测地可定向曲面。
On knot complements that decompose into regular ideal dodecahedra
Aitchison和Rubinstein构造了两个纽结补，它们可以分解为两个正则理想十二面体。本文证明了这些纽结补是唯一能分解成n个正则理想十二面体的纽结补，从而部分解决了Neumann和Reid的一个猜想。
Closed essential surfaces in the complements of large volume Berge knots
我们构造了一个算法，列出了所有封闭的本质表面的补充结，在于纤维的三叶草或图8结。这种结是伯格结，因此允许透镜空间手术。此外，它们可以具有任意大的双曲线体积。使用该算法，我们构造了两种类型的大体积Berge结：那些其补包含任意多个不同的封闭本质曲面的结，以及那些其补不包含封闭本质曲面的结。
Closed incompressible surfaces in the complements of positive knots
证明了任何闭的不可压缩曲面在正纽结的补空间中是代数上不可分裂的，正纽结不能束缚非自由的不可压缩Seifert曲面，正纽结和环的可分裂性和素性可以从它们的正图中看出.
We show that Bordered Heegaard Floer invariant $\widehat{CFD}$ of a knot complement in $S^3$ is invariant under the elliptic involution on its boundary.
Upper bounds for the complexity of torus knot complements
我们建立了具有非空边界的Seifert纤维流形复杂性的上界。特别是，我们得到了潜在的尖锐的界限上的复杂性环面结补。
Closed Quasi-Fuchsian Surfaces In Hyperbolic Knot Complements
我们证明了每个双曲纽结补包含一个封闭的拟Fuchsian曲面。
Closed incompressible surfaces of genus two in 3-bridge knot complements
本文刻画了亏格为2的闭不可压缩曲面在3-桥纽结和环的补中的性质。这个特征包括3桥结和链接的基本2弦缠结分解。
We establish a pair of criteria for proving that most knot complements obtained as Dehn fillings of a given two-component hyperbolic link complement lack hidden symmetries. To do this, we use certain rational functions on varieties associated to the link. We apply our criteria to show that among certain infinite families of knot complements, all but finitely many members lack hidden symmetries.
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.
In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism space between two type-DD modules, and derive type-DA morphisms from DD morphisms to derive the direct system maps. In addition, we introduce a direct limit invariant from the direct system which can detect non-unstable chains in the type-D module of a knot complement, if the type-…
Non-parallel essential surfaces in knot complements
我们表明，如果一个结或链接有n薄的水平时，把薄的位置，那么它的外部包含一个集合的n个不相交的，非平行的，平面的，非正交的，基本的表面。一个推论是，有至少n/3四面体在任何三角形的补充，这样一个结。
Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$.
Noncyclic covers of knot complements
亨佩尔证明了纽结补的基本群是剩余有限的。这意味着每一个非平凡的纽结必须有一个有限单的非循环覆盖。我们给出了一个明确的界限，$Φ(c)$，这样，如果$K$是一个非平凡的结在三个领域的图与$c$交叉和一个特别简单的JSJ分解，然后补充$K$有一个有限的，非循环覆盖至多$Φ(c)$表。
We develop a word mechanism applied in knot and link diagrams for the illustration of a diagrammatic property. We also give a necessary condition for determining incompressible and pairwise incompressible surfaces, that are embedded in knot or link complements. Finally, we give a finiteness theorem and an upper bound on the Euler characteristic of such surfaces.
Small knot complements, exceptional surgeries, and hidden symmetries
本文给出了$S^3$中小纽结补允许隐藏对称性的两个障碍。第一个障碍是与另一个结互补循环可交换。这一结果提供了一个部分的答案Boileau，Boyer，Cebanu，和沃尔什的猜想。我们还提供了第二个障碍，承认隐藏的对称性的情况下，一个小的结补涵盖了流形承认一些对称性和至少两个特殊的手术。
Neumann and Reid conjecture that there are exactly three knot complements which admit hidden symmetries. This paper establishes several results that provide evidence for the conjecture. Our main technical tools provide obstructions to having infinitely many fillings of a cusped manifold produce knot complements admitting hidden symmetries. Applying these tools, we show for any two-bridge link complement, at most finitely many fillings of one cusp can be covered by knot complements admitting hidden symmetries. We also show that the figure-eight knot complement is the unique knot complement with…
Great circle links and virtually fibered knots
我们表明，所有的两桥结和链接互补几乎纤维化。我们还表明，球形Montesinos结和链接补充几乎纤维。这是通过证明这样的结补语被大循环链补语覆盖来实现的。
The knot meridians of (1,1)-knot complements are CTF-detected
本文在$S^3$或透镜空间中非素(1,1)-纽结的余部构造了余定向紧叶状结构，证明其能探测纽结经线，为L-space猜想相关的Boyer-Gordon-Hu猜想提供了新证据。
Branched Spherical CR structures on the complement of the figure eight knot
在给定的完整表示下，我们得到了8字形纽结的补集上的分支球形CR结构（称为rho_2）。从8字形纽结的补到PU（2，1）的边界幂幺表示基本上有两个，我们称之为rho_1和rho_2。我们明确了这两种表示之间的一些根本区别。例如，将8字形纽结补看作圆上的一个面丛，在这种表示下，纤维的基本群的行为是rho_1和rho_2之间的一个关键区别。
It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirting…
Knots in circle bundles are determined by their complements
本文证明了亏格 $g \geq 2$ 曲面上的可定向圆丛中的纽结由其补集决定，解决了有向纽结补集猜想的一个情形，并应用于单位切丛和射影切丛中的典范纽结，给出了其补集同胚的充分必要条件。
It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.
In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let $M$ be a homology lens space with $H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$ and $K$ a not null-homologous knot in $M$. We show that $K$ is determined by its complement if $M$ is non-hyperbolic, $K$ is hyperbolic, and $p$ is a prime more than 7, or, if $M$ is actually a lens space $L(p,q)$ and $K$ represents a generator of $H_1(L(p,q))$.
Weights of essential surfaces in 2-bridge knot complements
通过分析结补集特征方程的理想点，研究2-桥结的本质曲面结构，并基于结图直接确定Serre树，推导出与不可压缩曲面相关理想点数量的公式。
Homotopy, Delta-equivalence and concordance for knots in the complement of a trivial link
链同伦和自Delta等价是链上的等价关系。这是由J.米尔诺（resp。Milnor不变量决定了一个链接是否是链接同伦的（分别是自Delta等价）到平凡链接。研究了环的某个分支上的环同伦和自Delta等价，即在环的补上的纽结的同伦和Delta等价。我们证明了Milnor不变量决定了平凡环的补链中的纽结是否是零同伦的，并给出了这种纽结与平凡纽结是Delta等价的充分条件.我们还给出了平凡纽结的补中的纽结等价到Delta-等价和协调的一个充分条件。
In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology $\mathbb{R}P^3$'s are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knot…
This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution on the knot Floer homology of $K$ which corresponds to moving the basepoints by one full twist around $K$ is trivial. The correction implicitly involves considering the contribution from this (possibly non-trivial) involution in a number of places.
Totally Geodesic Seifert Surfaces in Hyperbolic Knot and Link Complements II
我们推广了[AS]的结果，发现了大类的全测地塞弗特曲面在双曲纽结和链补，每个提升的刚性2-orbifold嵌入一些双曲3-orbifold。此外，我们提供了一个唯一性定理，并证明了许多结不能拥有完全测地塞弗特表面的宽度不变的存在下，这样一个表面的界限。最后，我们利用这些例子来证明，六定理是尖锐的纽结补的3-球面。
On the potential functions for the hyperbolic structures of a knot complement
我们解释了如何构造纽结补的双曲结构的某些势函数，这些势函数与双曲结构的变形空间上的解析函数密切相关。
This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a $S^2(2,3,6)$ cusp, it also covers an orbifold with a $S^2(3,3,3)$ cusp. We end with a discussion that shows all cusp types arise in the quotients of link complements.
Simplicial structures of knot complements
最近的研究表明，存在一个显式的Pachner移动的数量需要连接任何两个三角剖分的任何哈肯3-流形，其中不包含作为强简单的JSJ-分解的块被包围的子流形。在本文中，我们证明了一个推广的结果，所有结补。边界的显式公式是在两个三角剖分的四面体的数量。这给出了一个概念上平凡的算法，用于识别所有3-流形中的任何结补。
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}
In this article, we investigate the problem of counting totally geodesic surfaces in the complement of hyperbolic knots with at most 9 crossings. Adapting previous counting techniques of boundary slope and intersection, we establish uniqueness of a totally geodesic surface for the knots $7_4$ and $9_{35}$. Extending an obstruction to the existence of totally geodesic surfaces due to Calegari, we show that there is no totally geodesic surface in the complement of 47 knots.
In this paper we look at the knot complement problem for L-space $\mathbb{Z}$-homology spheres. We show that an L-space $\mathbb{Z}$-homology sphere $Y$ cannot be obtained as a non-trivial surgery along a knot $K\subset Y$. As a consequence, we prove that knots in an L-space $\mathbb{Z}$-homology sphere are determined by their complements.
Totally geodesic boundaries of knot complements
给定一个紧致可定向三维流形M，其边界是一个双曲曲面，在其边界上有一条简单闭曲线C，M中的每个纽结同伦于一个纽结，该纽结的补集具有一个完全双曲结构，其边界是全测地的，其中C的测地表示尽可能小。
Taut foliations in knot complements
我们证明了对于$S^3$中的任何非平凡结，存在一个包含零的开区间，使得在该区间中的任何斜率上的Dehn手术产生一个具有拉紧叶理的3-流形。这推广了Gabai关于零框架手术的一个定理。
Floer homology and knot complements
利用弗洛尔同调的Ozsvath-Szabo理论定义了三流形中纽结补的一个不变量。这个不变量采用过滤链复形的形式，我们称之为CF_r。它携带关于结上大型整体手术的弗洛尔同调的信息。使用确切的三角形，我们得到有关其他手术结的信息，以及由某些手术配边引起的弗洛尔同调的映射。我们在S^3中定义了一类\em{perfect}纽结，它的CF_r具有特别简单的形式。对于这些纽结，Ozsvath-Szabo理论的形式性质使我们能够对弗洛尔同调进行完整的计算。这是作者的论文;许多结果已经由Ozsvath和Szabo在Math.GT/0209056中独立发现。
Classical Knot Theory
本文是对纽结理论的一个非常简要的介绍。它描述了纽结着色的quandles，基本群的一个纽结补，和numerous-分解的纽结补。
We give a rational surgery formula for the Casson-Walker invariant of a 2-component link in $S^{3}$ which is a generalization of Matveev-Polyak's formula. As application, we give more examples of non-hyperbolic L-space $M$ such that knots in $M$ are determined by their complements. We also apply the result for the cosmetic crossing conjecture.
We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern-Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, Gelca and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a de…
Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links
设$F$是一个不可压缩的、非边界平行的曲面，其边界在代数缠结$(B,T)$的补曲面中.然后$F$分离$B$中的$T$串，并且$F$的边界斜率由$(B,T)$唯一确定，因此我们可以定义代数缠结的斜率.除了康威缠结和之外，我们还定义了两个缠结的自然积。代数缠结上的斜率和二元运算导致一种与有理数同构的代数结构。我们引入了一个新的纽结和链环类，代数交替的纽结和链环，粗略地说，这是从交替的纽结和链环通过替换一些交叉代数缠结。本文给出了代数交错纽结或链环$K$的补空间中闭曲面不可压缩和局部不可压缩的充要条件，特别证明了如果$K$是纽结，则$K$的补空间不包含这样的曲面.
Meridional Almost Normal Surfaces in Knot Complements
设$K$是闭3-流形$M$中的一个纽结，使得$\bar{M-N(K)}$不可约.我们发现，对于任何正整数$b$存在一个三角的$\bar{M-N(K)}$，使任何弱不可压缩的桥表面$K$的$b$桥梁或更少的是同位素的几乎正常的桥梁表面。
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely many geometric triangulations of a cusped hyperbolic 3-manifold. In contrast, our approach does not extend to the figure eight sister manifold, and it is unknown if there are infinitely many geometric triangulations for this manifold.
Nullhomotopic knots are determined by their complements
本文证明在基本群满足特定代数条件的三维流形中，零伦纽结由补空间唯一确定，从而回答了 Boileau 提出的 Kirby 问题，方法基于瞬子 Floer 同调与 SU(2)-表示簇。
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