We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds.
As a consequence, there are finitely many smooth closed 4-manifolds constructed from some Kirby diagram with bounded number of crossings, discs, and strands, or from some Turaev special shadow with bounded number of vertices. (These are the 4-dimensional analogues of Heegaard diagrams and special spines for 3-manifolds.) We therefore get two filtrations on the set of all smooth closed 4-manifolds with finite sets. The two filtrations are equivalent after linear rescalings, and their cardinality grows at least as n^{c*n}.
In this paper we study colored triangulations of compact PL 4-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of {\it special handlebody decompositions} for any simply-connected closed PL 4-manifold. In particular, we detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connecte…
A quandle is an algebra with two binary operations satisfying three conditions which are related to Reidemeister moves in knot theory. In this paper we introduce the notion of the (canonical) tensor product of a quandle. The tensor product of the knot quandle or the knot symmetric quandle of a surface-link in $4$-space can be used to classify or construct invariants of $1$-handles attaching to the surface-link.
A universal invariant of four-dimensional 2-handlebodies and three-manifolds
本文是我们关于四维2-三体工作的第二部分。在第一部分（arXiv：math.GT/0407032）中，我们证明了直到一定的局部移动集合，B^4在带状表面上分支的连通简单覆盖，双射表示连通的可定向4维2-D体直到2-变形（句柄滑动和索引<= 2的句柄的创建/取消）。我们通过一个映射将这种双射对应分解到由一个霍普夫代数对象H自由生成的泛辫子范畴中的闭态射上。用这种方法我们得到了四维2-三体的一个完整的代数描述。这个结果然后被用来获得类似的描述，这样的曲面体的边界，即3维流形，解决了闭流形的问题，提出了由Kerler在“Towards an algebraic characterization of 3-dimensional cobordisms”，Contemporary Mathematics 318（2003）。（参见：问题8-16（1）在T. Ohtsuki，“Problems on invariants of knots and 3-manifold”，Geom. Topol. Monogr. 4（2002年）。
Developed from geometric arguments for bounding the Morse-Novikov number of a link in terms of its tunnel number, we obtain upper and lower bounds on the handle number of a Heegaard splitting of a sutured manifold $(M,γ)$ in terms of the handle number of its decompositions along a surface representing a given 2nd homology class. Fixing the sutured structure $(M,γ)$, this leads us to develop the handle number function $h \colon H_2(M,\partial M;\mathbb{R}) \to \mathbb{N}$ which is bounded, constant on rays from the origin, and locally maximal. Furthermore, for an integral class $ξ$, $h(ξ)=0$ if…
Knot surgery $4$-manifolds $E(n)_K$ without $1$- and $3$-handles
Research institute of Mathematics, Seoul National University（首尔大学数学研究所）；School of Mathematics, Statistics and Data science, Sungshin Women’s University（诚信女子大学数学、统计与数据科学学院）
Spherical alterations of handles: embedding the manifold plus construction
在我们早期关于流形端点的工作中，一个关键的工具是能够将Quillen Plus Construction提供的协边嵌入到这些端点中。在这里，我们开发了一个“球形修改”的技巧，提供了一个建设性的方法来获得这样的嵌入。更重要的是，这种方法允许更一般的嵌入结果。在本文中，我们开发的推广，加上建设和相应的协边如何可以嵌入到流形满足适当的基本群性质。这里得到的结果在我们正在进行的非紧流形的研究中起着重要的作用。
Round handles, logarithmic transforms, and smooth 4-manifolds
圆柄在两个主要方面与光滑四维流形有联系：5维圆柄广泛地作为四维流形之间的协边的构建块出现，而四维圆柄是它们上的破莱夫谢茨纤维化的构建块。这篇文章的目的是为了更好地阐明这些相互作用。证明了：如果X和X'是具有相同欧拉示性数的共边闭光滑4-流形，且其中一个是单连通的，则它们之间存在一个仅由圆2-柄构成的共边，从而可以通过一系列沿环面的广义对数变换从一个到另一个.作为推论，我们得到了岩濑的一个定理的新证明，它是三维流形的Lickorish-Wallace定理的四维类似：每个闭的单连通4-流形可以通过沿着包含在CP ^2，-CP ^2和S^1 x S^3的连通和中的环面的不相交并的手术产生。这些回答了Ron Stern提出的一些开放性问题，同时建议考虑对配边法进行更多的限制。我们还使用圆柄来证明，在迄今为止构造的同一同胚类中，每个相互非同构的闭光滑定向单连通4-流形的无限族都包含在一次稳定化之后成为同构的成员，如果成员都是非自旋的，则为S^2 x S^2，如果成员都是自旋的，则为S^2 x S^2 # -CP^2。特别地，我们证明了简单的协边存在于结涌流形之间。然后，我们表明，广义对数变换可以被看作是标准的对数变换沿纤维组件的破莱夫谢茨纤维化，并显示如何改变光滑结构上的一个固定的同胚类的一个封闭的光滑4流形可以实现相关的修改破莱夫谢茨纤维化。
The construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.
Let $M$ be a simple 3-manifold, and $F$ be a component of $\partial M$ of genus at least 2. Let $α$ and $β$ be separating slopes on $F$. Let $M(α)$ (resp. $M(β)$) be the manifold obtained by adding a 2-handle along $α$ (resp. $β$). If $M(α)$ and $M(β)$ are $\partial$-reducible, then the minimal geometric intersection number of $α$ and $β$ is at most 8.
Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in which the adjacency relation on handles may give an arbitrary graph. We show that for compact hyperbolic 3-manifolds, this is linearly related to a notion of metric complexity, based on the areas of level sets of Morse functions to graphs, which we call Gromov area.
It is well-known that in dimension 4 any framed link $(L,c)$ uniquely represents the PL 4-manifold $M^4(L,c)$ obtained from $\mathbb D^4$ by adding 2-handles along $(L,c)$. Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram $(L^{(*)},d)$), the associated PL 4-manifold $M^4(L^{(*)},d)$ is obtained from $\mathbb D^4$ by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored …
We introduce and define "oriented framed measured lamination links" in a 3-manifold $M$. These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links bound 2-dimensional measured and oriented "Seifert laminations." We show that any lamination link which bounds a 2-dimensional Seifert lamination, bounds a "taut" Seifert lamination, i.e. one of maximum Euler characteristic, subject to the condition that the Seifert lamination is car…
M. Freedman and V. Krushkal introduced the notion of a "filling" link in a 3-manifold: a link $L$ is filling in $M$ if for any spine $G$ of $M$ disjoint from $L$, $π_1(G)$ injects into $π_1(M \setminus L )$. Freedman and Krushkal show that there exist links in the $3$-torus $T^3$ that satisfy a weaker form of filling, but they leave open the question of whether $T^3$ contains an actual filling link. We answer this question affirmatively by proving in fact that every closed, orientable 3-manifold $M$ with $\operatorname{rank}(π_1(M)) = 3$ contains a filling link.
Let $M_1$ and $M_2$ be closed connected orientable $3$-manifolds. We classify the sets of smooth and piecewise linear isotopy classes of embeddings $M_1\sqcup M_2\rightarrow S^6$.
We calculate the volume of the $7_3^2$ link cone-manifolds using the Schläfli formula. As an application, we give the volume of the cyclic coverings over the link.
We study the set $\widehat{\mathcal S}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold $M$. We show that $\widehat{\mathcal S}_M$ is well-defined and describe how it relates to exotic phenomena in dimension four. In particular, in the case when $X$ is smooth, with a handle decompositions with no 1-handles and homeomorphic to but not smoothly embeddable in $D^4$, our results tell us that $X$ is exotic if and only if there is a link $L\hookrightarrow S^3$ which is smoothly slice in $X$, but not in $D…
Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4-manifolds with multiple boundary components, and show how trisected 4-manifolds with multiple boundary components glue together.
Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear $n$-manifold has a multisection, i.e. can be divided into $k+1$ $n$-dimensional $1$-handlebodies, where $n=2k+1$ or $n=2k$, such that intersections of the handlebodies have spines of small dimensions. Several applications, constructions and generalisations of our approach are given.
We show that three natural decision problems about links and 3-manifolds are computationally hard, assuming some conjectures in complexity theory. The first problem is determining whether a link in the 3-sphere bounds a Seifert surface with Thurston norm at most a given integer; this is shown to be NP-complete. The second problem is the homeomorphism problem for closed 3-manifolds; this is shown to be at least as hard as the graph isomorphism problem. The third problem is determining whether a given link in the 3-sphere is a sublink of another given link; this is shown to be NP-hard.
We present the Round Handle Problem, proposed by Freedman and Krushkal. It asks whether a collection of links, which contains the Generalised Borromean Rings, are slice in a 4-manifold R constructed from adding round handles to the four ball. A negative answer would contradict the union of the surgery conjecture and the s-cobordism conjecture for 4-manifolds with free fundamental group.
