Sergey A. Melikhov
We construct a link in the $3$-space that is not isotopic to any PL link (non-ambiently). In fact, there exist uncountably many $I$-equivalence classes of links. The paper also includes some observations on Cochran's invariants $β_i$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (resp. contractible open 3-manifold) M if and only if M is homeomorphic to S^3 (resp. R^3). As a byproduct of the proof of the "if" part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M. We show that k-quasi-i…
Smoothing topological isotopy of surfaces in 4-manifolds with a boundary geometric dual
本文证明在带边界几何对偶的光滑可定向4流形中，整洁曲面的拓扑同痕蕴含光滑同痕，从而建立光滑与拓扑同痕类间的双射。
In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant β^k, provided k>2, and a series of \barμ-invariants, starting with \barμ(111112122) for k=3, also fall in this gap. In fact, all \barμ-invariants where each index occurs at most k+1 times, except perhaps for one occuring k+2 times, can be extracted from G_k, and if they vanish, G…
Two-variable Conway polynomial and Cochran's derived invariants
本文研究链环的双变量Conway多项式与Cochran导出不变量的关系，在环绕数为0和1两种情形下给出系数$α_{1,2k-1}$的公式，并用Seifert矩阵提供新证明。
In this paper we define and study flexible links and flexible isotopy in projective space. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholms interpretation of Viros encomplexed writhe.
We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a long-standing question as to whether certain natural families of knots contain topologically slice knots. We also present a simpler proof of the result of Cochran-Teichner that the successive quotients of the integral terms of the Cochran-Orr-Teichner filtration of the knot concordance group have rank 1. For links we have similar results. We show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence…
Topological isotopy and finite type invariants
研究证明有限型不变量能够区分$S^3$中的PL链接当且仅当Rolfen问题有肯定答案，并讨论了相关不变量的性质。
New examples of topologically slice links
本文提出了一个新的 Milnor 不变量条件，用于处理原始定理无法覆盖的多重感染链环拓扑切片情况，并给出了多重感染为 $(n-0.5)$-可解的充分条件。
Light bulb smoothing for topological surfaces in 4-manifolds
本文提出4-流形中拓扑嵌入曲面的新光滑化技术，实现拓扑到光滑的同痕，并应用于证明维度4中某些圆盘和球面的“拓扑=光滑”结果，同时获得Dax不变量的拓扑版本。
Pseudo-isotopy versus isotopy for homeomorphisms of 4-manifolds
该研究定义了四维流形上拓扑伪同痕无法同痕于同痕的阻碍，其与Hatcher-Wagoner光滑阻碍匹配且取值于Whitehead群，证明阻碍可完全实现，并构造出与恒同映射伪同痕但不同痕的Y×S¹同胚。
Geometric Constraints in Link Isotopy
本文通过参数化厚度证明存在几何锁定的拓扑平凡纽结（戈尔迪安平凡纽结），并揭示曲率与自距离约束将平凡纽结合痕类分裂为无穷多个不连通分量的分层结构。
Smooth stable isotopy of topologically isotopic surfaces
研究在4-流形中拓扑同痕的曲面是否在某个稳定化（与若干$S^2\times S^2$做连通和）后光滑同痕，证明当曲面在$\mathbb{Z}/2$-同调中平凡时成立，并给出成立的一大类基本群（包括经典纽结群的自由积和自由群）。
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In both of these cases, each knot space consists of five components, but contains only three (when n = 6) or four (when n = 7) topological knot types. Therefore ``geometric knot equivalence'' is strictly stronger than topological equivalence. This point is demonstrated by the hexag…
Creating Stein surfaces by topological isotopy
本文结合 Freedman 拓扑与 Eliashberg 全纯理论，证明复曲面中驯服拓扑嵌入的 2-复形经小拓扑同位素后可作为不可数多个 Stein 正则邻域的交集，并借此研究拓扑伪凸性与伪凹性。
Pseudo-isotopies of simply connected 4-manifolds
本文通过绕过替换准则，完善了Quinn关于单连通紧致4维流形拓扑和光滑伪同胚定理的证明，并讨论了替换准则的正确性问题。
We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.
The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of deeper structures in the smooth concordance group of topologically slice knots. We show that the graded quotient of the bipolar filtration of topologically slice knots has infinite rank at each stage greater than one. To detect nontrivial elements in the quotient, the proof simultaneously uses higher order amenable Cheeger-Gromov $L^2$ $ρ$-invariants and infinitely many Heegaard Floer correction term $d$-invariants.
It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for some (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C^0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k=1,2,...; all of the…
Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be split with a physical isotopy.
We use a link invariant defined by Cimasoni-Florens to compute ρ-invariants. This generalizes results of Cochran-Teichner and Friedl on knots to the setting of links. As an application, we prove with only twelve possible exceptions that the twist knots of algebraic order two are linearly independent in the topological concordance group.
We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with $S^2 \times S^2$, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curvature for such 4-manifolds. We also construct families of smoothly distinct links, all of whose corresponding proper sublinks are smoothly isotopic, that become smoothly isotopic after stabilizing.
For Gamma a finite, connected metric graph, we consider the space of configurations of n points in Gamma with a restraint parameter r dictating the minimum distance allowed between each pair of points. These restricted configuration spaces come up naturally in topological robotics. In this paper, we study the homotopy, homeomorphism, and isotopy types of these spaces over the space of parameters r and provide a polynomial upper bound (in the number of edges of Gamma) for the number of isotopy types.
In 2016 Levine showed that there exists a knot in a homology 3-sphere which is not smoothly concordant to any knot in the 3-sphere where one allows concordances in any smooth homology cobordism. Whether the same is true if one allows topological concordances is not known. One might hope that such an example might be detected by the powerful filtration of knot concordance introduced by Cochran-Orr-Teichner. We prove that this is not the case, demonstrating that for any knot in any homology sphere there is a knot in the 3-sphere equivalent to the original knot modulo any term of this filtration.…
We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to the boundary, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M,v,L), with (M,v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transversality-preserving simultaneous homotopy…
The bipolar filtration introduced by T. Cochran, S. Harvey, and P. Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1-bipolar knots which are not 2-bipolar. For knots, this is the highest known level at which the filtration does not stabilize. For the case of links with two or more components, we prove that the filtration does not stabilize at any level: for any n, there are topologically slice links which are n-bipolar but not (n+1)-bipolar. In the proof we describe an explicit geometric construction…
We establish certain "non-triviality" results for several filtrations of the smooth and topological knot concordance groups. First, as regards the n-solvable filtration of the topological knot concordance group defined by K. Orr, P. Teichner and the first author, we refine the recent non-triviality results of Cochran and Teichner by including information on the Alexander modules. These results also extend those of C. Livingston and the second author. We exhibit similar structure in the closely related symmetric Grope filtration of the knot concordance group considered by the first author and T…
This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way. Let $M$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M_n$ be the moduli space of all n-cables $nK$ of framed long knots $K$ which are twisted by a given string link $T$ to close to a knot in the solid torus, with a marked point on the knot at infinity. First we construct integer valued combinat…
We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the problem by Kawamuro.
Thin Gordian Unlinks
本文首次构造出戈尔迪安解链的例子，并引入厚度在[1,2)的单参数细纽结模型，证明更细的管状邻域会产生不同于经典粗纽结的绳结几何。
For a classical link, Milnor defined a family of isotopy invariants, called Milnor $\overlineμ$-invariants. Recently, Chrisman extended Milnor $\overlineμ$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $\overlineμ$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $\overlineμ$-invariants of classical links.
This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction th…
The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.
Isotopy of the Dehn twist on K3#K3 after a single stabilization
本文利用 Pin(2)-等变 Bauer-Furuta 不变量，证明了 $K3\#K3$ 颈区的 Dehn 扭转在单次稳定化后仍不光滑同痕于恒等映射，给出了单连通光滑 4-流形上单次稳定化后不消失的奇异现象的首个例子。
Learning Topological Invariance
提出自动化学习拓扑不变性的机器学习方法，将其应用于纽结理论，通过对比与生成技术映射同一纽结类表示，还开发师生框架解释学习的不变量，并探索解决琼斯无结猜想的方法。
In the prequel of this paper, Kauffman and Ogasa introduced new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$ where $I$ denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are new, nontrivial, and calculable examples of quantum in…
We introduce a technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. We define generalized doubling operators, of which Bing doubling is an instance, and prove our nontriviality results in this more general context. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group.
On The Cost Function Associated With Legendrian Knots
本文提出衡量拓扑同痕勒让德纽结转化为勒让德同痕障碍的 Cost 函数，证明其度量性质，并用于刻画、定量稳定化定理及多类纽结计算。
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3-spheres which are indistinguishable via multisignatures, eta-invariants, and L2-signatures but have distinct hom…
Operadic actions on long knots and 2-string links
本文将2-弦链空间实现为标色算子$\mathcal{SCL}$上的自由代数，扩展了前人关于商空间和长结自由性的结果，并从代数和拓扑角度细化了弦链同痕类幺半群的定理。
Seifert surfaces in the 4-ball
本研究利用Khovanov同调的配边映射，构造出三维球面中同一纽结的同亏格塞弗特曲面推入四维球后不同痕的例子，解答了Livingston1982年的问题，并验证了配边映射在卫星运算下的稳定性与可计算性。
Polynomial algorithm for alternating link equivalence
本文研究两个交替链环图的同痕等价判定，结合 Tait flyping 猜想及图论复杂度结果，给出以最大交叉点数计的多项式时间算法。
The crossing and the arc from the topological viewpoint
本文从拓扑视角研究纽结图元素，将弧和交叉点的等价类描述为探针的同痕类，探讨其同伦类在曲面tangle着色代数结构中的基础作用，并引入multicrossing complex统一多种同调理论。
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to diagrams of the initial object. The main tool is the notion of {\em free $k$-braid group}. In the simplest case, for free $2$-braids, the word problem and the conjugacy problem can be solved by finding the minimal representative, which can be thought of as a graph, and is unique, as such. We prove a general theorem about invariants of dynamical systems which ar…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。