W. B. R. Lickorish
Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some contractible 4-manifold in the 4-sphere.
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Lickorish has constructed large families of contractible 4--manifolds that have knotted embeddings in the 4--sphere and has also shown that every finitely presented perfect group with balanced presentation occurs as the fundamental group of the complement of a knotted contractible manifold. Here we make a few observations regarding Lickorish's construction, showing how to extend it to construct contractible 4--manifolds which have an infinite number of knotted embeddings and also to construct knotted embeddings of the Mazur manifold for which the complement has trivial fundamental group.
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。
We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots $6_{14}$ and $6_{15}$ are not equivalent. We also show that some genus two handlebody-knots with a knotted handle decomposing sphere can be determined by their exteriors. As an application, we show that the exteriors of $6_{14}$ and $6_{15}$ are not homeomorphic.
Given a properly embedded graph Gamma in a ball B and a punctured sphere Sigma properly embedded in B - Gamma, we examine the conditions on Gamma that are necessary to assure that Sigma is boundary parallel.
In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization of a knotted sphere and prove that a (+/-)-twist spin of a knotted trivalent graph may be knotted. We then construct some families of knotted graphs for which the (+/-)-twist spins are always unknotted.
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。
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.
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…
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。
We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and …
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, …
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。
Exotically knotted 2-spheres and the fundamental groups of their complements
该研究展示了一种构造包含特定 2 链的 4 维 manifold 的方法，其 2 链群可为任意有限生成群，并证明这些链的组成部分具有空同源性质。
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…
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。
$n$-knots in $S^n\times S^2$ and contractible $(n+3)$-manifolds
本研究推广Mazur的经典构造，对任意整数$n\geq2$，通过构造$S^n\times S^2$中推广Mazur模式的纽结化$n$-球面，利用0-把柄、$n$-把柄和$(n+1)$-把柄得到与标准$(n+3)$-球体不同胚的带边界可缩紧致光滑$(n+3)$-流形。
Knotted handlebodies in the 4-sphere and 5-ball
本文构造了$S^4$中具有相同边界与切割系统但不同痕的3维1-柄体，证明了Budney-Gabai关于亏格至少2的猜想。
Classical and new plumbed homology spheres bounding contractible manifolds and homology balls
本文研究哪些铅垂同调3-球面界定可缩4-流形或同调4-球，通过修改Mazur论证并利用Poénaru流形，重新证明经典例子并给出两个新的无穷族。
Welded graphs, Wirtinger groups and knotted punctured spheres
本文建立焊接图的图示理论并扩展 Satoh 的 Tube 映射，用以分类纽结穿孔球面、刻画其 Wirtinger 群，并提出保持 Tube 映射不变的 Υ 局部移动。
The paper [10] incorrectly asserts that the digital image MSS_18, a digital model of the Euclidean 2-sphere S^2, is not 18-contractible. We show this assertion is false.
We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in a compact orientable 3-manifold is equal to a given integer is in NP. As a corollary, the problem of determining the genus of a knot in the 3-sphere is in NP. We also show that the problem of determining whether a compact orientable 3-manifold has incompressible boundary is in NP.
Knotting and linking in 4 and 5 dimensions from barbell diffeomorphisms
本文通过双耳微分同胚构造了无限多个非同胚的3结，每个具有四个临界点，反驳了Scharlemann定理，并解决了Budney和Gabai关于存在结状处理体的猜想最后情况。
We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of $S^2 \times S^1$, then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact orientable surface to be contractible, which generalizes the result by Hatcher that the arc complexes are contractible. We then construct infinitely many Heegaard splittings, including the above-mentioned Heegaard splitting, for which suitably defined complexes of Haken spheres a…
For every N > 0 there exists a group of deficiency less than -N that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold of the n-sphere, n > 4, if and only if it is the fundamental group of a homology n-sphere. There exist fundamental groups of homology n-spheres, n > 4, that cannot arise as the fundamental group of the complement of a contractible submanifold of the 4-sphere.
Let G be a graph in a 3-manifold M. We compress the pair (M,G) along admissible 2-spheres as long as possible. What we get is a root of (M,G). Our main result is that for any pair (M,G) the root exists and is unique. As a corollary we get an easy proof of Petronio's theorem on prime decompositions of 3-orbifolds.
It is known that every exotic smooth structure on a simply connected closed 4-manifold is determined by a codimention zero compact contractible Stein submanifold and an involution on its boundary. Such a pair is called a cork. In this paper, we construct infinitely many knotted imbeddings of corks in 4-manifolds such that they induce infinitely many different exotic smooth structures. We also show that we can imbed an arbitrary finite number of corks disjointly into 4-manifolds, so that the corresponding involutions on the boundary of the contractible 4-manifolds give mutually different exotic…
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。
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。
In this paper we study embeddings of oriented connected closed surfaces in $\mathbb S^3$. We define a complete invariant, the fundamental span, for such embeddings, generalizing the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived and employed to study handlebody knots, bi-knotted surfaces, and chirality of knots. These invariants are capable to distinguish inequivalent handlebody knots and bi-knotted surfaces with homeomorphic complements. Particularly, we obtain an alternative proof of the inequivalence of Ishii et al.'s ha…
Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds
本文利用不等价光滑结构，在4-流形中构造无穷多光滑互异但拓扑同伦的零同调曲面，并指出其拓扑边界为柄体。
There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which the actions extend locally linearly but not smoothly.
Pants distances of knotted surfaces in 4-manifolds
定义4-流形中纽结曲面的裤子距离，推广了Blair-Campisi-Taylor-Tomova对4-球面中曲面的复杂度研究，并利用该距离刻画了标准形式和计算了多个新例子的精确值。
In this note, we investigate genera for the slopes of a knotted torus in the 4-sphere analogous to the genus of a classical knot. We compare various formulations of this notion, and use this notion to study the extendable subgroup of the mapping class group of the knotted torus.
A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in ${\mathbb C}^2$, with either orientation. A related question asks whether every compact contractible 4-manifold admits the structure of a Stein domain. We verify Gompf's conjecture, with one orientation, for a family of Brieskorn spheres of which some are known to admit a smooth embedding in ${\mathbb C}^2$. With the other orientation our methods do not resolve the question, but do give rise to an example of a contractible, boundary-irreducible 4-manifold that admits no Stein struc…
Exotically knotted closed surfaces from Donaldson's diagonalization for families
本研究利用族版本Donaldson对角化定理，提出无需显式用4-流形不变量的奇异曲面检测法，构造出首个补空间微分同胚的不可定向奇异纽结曲面例子，并得到多类奇异嵌入的相关结果。
A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \times S^1$ or the connected sum whose summands are lens spaces or $S^2 \times S^1$, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precis…
On homology planes and contractible $4$-manifolds
本文通过追溯Kirby的初始构造，首次给出Kirby-Ramanujam球面的新例子及三个无限族，并证明其中一个族微分同胚于两个Brieskorn球面族的拼接，揭示了其同调配边群中的平凡性与拼接分量线性无关性的对比。
Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical projection induces the branched covering map. In the case of the 3-sphere, the branch locus is a knotted or linked curve in space, the 2-fold branched cover always embeds, and the 3-fold branch cover might be immersed. In the case of the 4-sphere, the branch locus is a knotted or link…
Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in the (n+2)-space. There are two inequivalent embeddings of an n-manifold in the (n+2)-space such that the projection of one of these in (n+1)-space has no double points and the projection…
Here we study two interesting smooth contractible manifolds, whose boundaries have non-trivial mapping class groups. The first one is a non-Stein contractible manifold, such that every self diffeomorphism of its boundary extends inside; implying that this manifold can not be a loose cork. The second example is a Stein contractible manifold which is a cork, with an interesting cork automorphism $f:\partial W \to \partial W$. By \cite{am} we know that any homotopy $4$-sphere is obtained gluing together two contractible Stein manifolds along their common boundaries by a diffeomorphism. We use the…
The Alexander trick for homology spheres
本文证明在维度至少为6时，固定边界的紧致可缩流形同胚群可缩，并由此推导出单侧$h$-配边的强唯一性。
Spanning solids and Murasugi sum in dimension four
研究四维球面上打结曲面的跨越实体构造，借助村杉和的四维类似物定义树状打结曲面概念，通过给出例子及应用于破碎曲面图，确定交叉点分辨率以产生跨越实体。
We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic definitions are due to Evako: the unit sphere of a vertex x of a graph G=(V,E) is the graph generated by {y | (x,y) in E} Inductively, a finite simple graph is called contractible if there is a vertex x such that both its unit sphere S(x) as well as the graph generated by V-{x} are…
Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2 subgroup H of G, M_H = N/H is a double branched cover of M. Sakuma has proved that the first homology of N is isomorphic, modulo 2-torsion, to the direct sum of the first homology groups of the M_H, and has shown that H_1(N) is determined up to isomorphism by the direct sum of the H_1(M_H) in certain cases; specifically, when d=2 and the coloring is such that t…
It was proven in the first author's paper "Contact 3-manifolds twenty years since J. Martinet's work" (Ann. Inst. Fourier, 42(1992), 165--192) that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. It was also claimed there without a proof that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on $S^3$, fixed at a point, is contractible. We prove this result in the current paper.
Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the 2-torus we have the normal disk pair: a 4-dimensional disk and a 1-dimensional proper sub-disk. We consider an isotopy (deformation) of the normal 1-disk inside the normal 4-disk, by using a map from the 2-torus to the space of long knots in 4-space, first considered by Budney. We use this isotopy in a construction called spinning about a submanifold introdu…
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…
An exotic 5RP^2 in the 4-sphere
该研究构造了4维球面中拓扑标准但光滑打结的5RP^2嵌入实例，证明其与标准嵌入不光滑同痕，为低维拓扑中光滑与拓扑嵌入的差异提供了新例证。
We consider pairs (X,Y) where X is a compact, locally CAT(-1) space, and Y is a totally geodesic subspace. The inclusion induces an embedding of the boundaries at infinity of the universal covers; we focus on the case where these are spheres whose dimensions differ by 2. We show that if the embedding is tame, then it is unknotted. We give examples of pairs for which the embedding is knotted (and can be realized as the fixed point set of an involution). We also provide a criterion for knottedness of tame codimension two spheres in high dimensional (>5) spheres. Corresponding results for delta-h…
The $L^p$-diameter of the space of contractible loops
本文证明任意紧致可定向曲面上固定面积的可缩简单环空间在 $L^p$-度量下具有无限直径，并引入比拟态射更一般的新泛函解决辛拓扑中的相关问题。
On homology concordance in contractible manifolds and two bridge links
本文通过构造边界为光滑可缩4-流形的一族流形-纽结对，证明了同调配边群商群包含Z^∞-直和项，并利用Ozsváth-Szabó-Hales算法对一类纽结的Floer同调进行了分类。
Heegaard diagrams for $5$-manifolds
本文定义了$5$维配边、柄体和闭流形的Heegaard图，给出表示与移动等价定理，并用于研究Gluck扭转是否同胚于标准$4$球面。
Homotopy classification of knotted defects in ordered media
本文提出了一种基于同伦理论对有序介质中结状缺陷进行分类的方法，通过四元数群Q的子群结构和共轭类分析，实现了对缺陷的系统分类。
The main result of this paper is that any $3$-dimensional manifold with a finite group action is equivariantly, invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology $3$-sphere also acts on a hyperbolic homology $3$-sphere. The theorem has other applications, including establishing the existence of an infini…
Homeomorphisms of surfaces in $4$-manifolds
本文建立单连通4维流形中局部平坦扭结曲面等价的充要条件，推广Lee-Wilczynski结果，证明射影平面等的相关结论，确定拓扑可延拓映射类群。
Deeply Slice Knot Detection via Immersed Curves
本文通过构造一类深切片结，利用Heegaard Floer同调中的不变量和浸入曲线技术，对Kirby列表中Akbulut提出的关于同调3-球面的问题取得进展。
We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions.
Brieskorn spheres bound orbit spaces of torus manifolds
本文构造了边界为非平凡基本群Brieskorn同调3-球面的特定环面8-流形，并研究其向GKM图扩展的复杂度条件。
We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a $4$-manifold with boundary. Using the relative Whitney trick we prove that every link in a homology sphere is homotopic to a link that is topologically slice in a contractible topological $4$-manifold. We further prove that any link in a homology sphere is order $k$ Whitney tower concordant to a link in $S^3$ for all $k$. Finally, we explore the minimum Gordian distance from a link in $S^3$ to a homotopically trivial link. Extending this notio…
We show that 4-dimensional conjugation manifolds are all obtained from branched 2-fold coverings of knotted surfaces in Z/2-homology 4-spheres.
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