Greg Friedman
We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.
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
We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These invariants are invariants of concordance and of J-characteristicness when f induces the identity on the fundamental group of M, and may be viewed as extensions of Milnor's invariants. We do not require the knots considered here to be rationally null-homologous or framed.
We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this "integration" homotopy-theoretically, we are able to produce integral cohomology classes. We then show how this integration is compatible with the homology operations on the space of long knots, as studied by B…
Homotopy index polynomials for knotoids
定义了定向曲面中纽结样的同伦指标多项式及同伦类型，研究其基本性质，表明平面纽结样的同伦多项式不比之前定义的指标多项式弱，且能检测球面纽结样的不可旋转性。
This work continues the study of a homotopy-theoretic construction of the author inspired by the Bott-Taubes integrals. Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots. Their techniques were later used by Cattaneo et al. to construct real "Vassiliev-type" cohomology classes in spaces of knots in higher-dimensional Euclidean space. By doing this integration via a Pontrjagin-Thom construction, we constructed cohomology classes in the knot space with arbitrary coefficients. We later showed that a refinement of this …
We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.
Graphing, homotopy groups of spheres, and spaces of long links and knots
本文研究高余维长链环空间的同伦群，利用图示映射证明球面同伦群在广义无三重点范围内构成全部来源，并描述范围外长纽结首个非平凡同伦群的 Borromean 环与 Hopf 纤维化生成元。
By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introduce a complete knot invariant defined via the stratified homotopy groups ----- En considérant des homotopies préservant la stratification, on obtient une notion naturelle d'homotopie pour les espaces stratifiés. Dans cette note, on présente des invariants d'homotopie stratifiée, …
This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate' the components via the companionship trees associated to the knot. The knots with the simplest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy-type of these components of K were computed by Hatcher. In the case the companionship tree has height, we give a fibre-bundle description of those components of K, recursively, in…
There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sl_n complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To …
In this paper we construct some invariants of spatial graphs by disk-summing the constituent knots and show the delta edge-homotopy invariance of them. As an application, we show that there exist infinitely many slice spatial embeddings of a planar graph up to delta edge-homotopy, and there exist infinitely many boundary spatial embeddings of a planar graph up to delta edge-homotopy.
Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which relates the Morse-Novikov numbers of N and K and generalizes the classical results of D. Roseman and E.C. …
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition…
In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with this correspondence. This correspondence is given by a simple formula.
We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.
More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres
本文通过用无限族 Montesinos 纽结替代 pretzel 纽结作为输入，扩展了 Miyazawa 的奇异嵌入族，并证明相应 roll-spun 纽结的分支双覆盖为同伦球面，从而构造出更大的同伦球面与同伦复射影平面族。
Invariance and naturality of knot lattice homology and homotopy
该研究证明纽结格同调是有理同调球中广义代数纽结光滑型的不变量，可由双重滤过同伦型的胞腔同调实现，还揭示了其同伦的自然性及奇点分解拓扑的相关结论。
Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define $M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of $LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with this link, we construct homotopy $K3$ and spin rational homology $K3$ surfaces containing $M_n(K)$. Then we apply the adjunction inequality to show that if $n>2g^n_s(K)-2$, $M_n(K)$ does not bound any smooth spin rational $4$-ball, and that under the same assumption the negative $n$-twisted Whitehead double of $LHT \sharp K$ is not a slice knot, where $g^n_s(K)$ is th…
We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs of Milnor's theorem on free groups for these theories and their pointed variants, identifying the homotopy types of the free racks and free quandles on spaces of generators. These results allow us to complete the stable classification of racks and quandles by identifying the ring spectra that model their stable homotopy theories. As an application, we show that the stable homotopy of a knot quandle is, in general, mor…
Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots
针对Plotnick--Suciu构造及Suciu的替代构造，研究发现基本quandle等对象无法决定定向2-纽结外部空间的同伦类型，还明确了Tanaka--Taniguchi例子的二阶同伦模性质。
Murasugi sums can be defined as readily for Morse maps to the circle of (arbitrary) link complements in the 3-sphere as for fibrations over the circle of (fibered) link complements in the 3-sphere. As one application, I show that if a knot K has free genus m, then there is a Morse map from its complement to the circle (representing the relative homology class of a Seifert surface for K) with no more than 4m critical points.
We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e. regular neighborhoods of embedded graphs, where a knot in a 3-subspace of $S^3$ is said to be transient if it can be moved by a homotopy within the subspace to the trivial knot in $S^3$. To show this, we discuss relation between certain group-theoretic and homotopic properties of knots in a comp…
Calegari's homotopy 4-spheres from fibered knots are standard
本文证明了 Calegari 于 2009 年由纤维化纽结单值群构造的光滑同伦 4-球均微分同胚于标准 4-球，利用了 5 维柄体技术与 3 维柄体映射类群结果，并提出了光滑 4 维 Schoenflies 猜想的潜在反例。
A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial knots in $\mathbb{R}^3$ with the inductive limit topology coming from the spaces $\mathcal{O}_d$ for $d\geq3$, where $\mathcal{O}_d$ is the space of polynomial knots in $\mathbb{R}^3$ with degree $d$ and having some conditions on the degrees of the component polynomials. In the same paper, we have proved that the space of polynomial knots in $\mathbb{R}^3$ has t…
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。
A knot Floer stable homotopy type
本文通过网格图构造过滤谱，其同调为纽结Floer同调，并猜想其过滤同伦型为纽结不变量，方法基于组合定义而非全纯几何。
We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant $v_2(K)$ by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we …
Homotopy type of stabilizers of circle-valued functions with non-isolated singularities on surfaces
本文研究了有向曲面上光滑函数稳定子的同伦性质，完整描述了一类Morse-Bott函数推广的光滑函数稳定子恒等映射连通分量的同伦类型。
A satellite formula for real Seiberg-Witten Floer homotopy types
本文建立了具有奇模式的纽结的实Seiberg-Witten Floer同伦类型的卫星公式，基于实Floer同伦类型的切除定理，导出了纽结配边理论的应用，并证明配边不变量仅依赖于零框架手术。
In this article, we study the knots realized by periodic orbits of R-covered Anosov flows in compact 3-manifolds. We show that if two orbits are freely homotopic then in fact they are isotopic. We show that lifts of periodic orbits to the universal cover are unknotted. When the manifold is atoroidal, we deduce some finer properties regarding the existence of embedded cylinders connecting two given homotopic orbits.
Assume $Γ$ is a negative-definite forest with exactly one unframed vertex, and $M(Γ)$ is the resulting plumbed 3-manifold with a knot embedded. We show that the filtered lattice chain homotopy type of $Γ$ is an invariant of the diffeomorphism type of $M(Γ)$.
Distinguishing 2-knots admitting circle actions by fundamental groups
本文研究3-球面中不同1-结与互质整数对构造的分支扭转旋转的唯一性问题，证明大多数情况下不存在产生相同分支扭转旋转的不同1-结，并利用循环型3-轨道流形的基本群作为补空间基本群的商群来区分分支扭转旋转。
In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3).
This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension.
We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine-Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knot…
We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$. We show that for many two-bridge knots, this cover contains a closed incompressible surface. Infinitely many fillings of the two-bridge knot lift to fillings of great circle link where the incompressibility of this surface is preserved. Using this, we show that infinitely many f…
An iterable surgery formula on involutive knot lattice homotopy
本文提出可迭代的纽结格点同调手术公式，生成对偶纽结的双重过滤空间及对合数据，并在无穷范畴中实现函子性，用于计算具体例子。
In 2001, Östlund formulated the question: are Reidemeister moves of types 1 and 3 sufficient to describe a homotopy from any generic immersion of a circle in a two-dimensional plane to an embedding of the circle? The positive answer to this question was treated as a conjecture (Östlund conjecture). In 2014, Hagge and Yazinski disproved the conjecture by showing the first counterexample with a minimal crossing number of 16. This example is naturally extended to counterexamples with given even minimal crossing numbers more than 14. This paper obtains the first counterexample with a minimal cross…
Circles Minimize most Knot Energies
该研究定义了一类新的重整化纽结能量，证明多数此类能量及O'Hara的多数纽结能量由正圆唯一最小化，证明了两个相关猜想，还找到非正圆最小化的能量，证明基于Luko的闭合曲线平均弦长定理。
Pseudodiagrams are knot or link diagrams where some of the crossing information is missing. Pseudoknots are equivalence classes of pseudodiagrams, where equivalence is generated by a natural set of Reidemeister moves. In this paper, we introduce a Gauss-diagrammatic theory for pseudoknots which gives rise to the notion of a virtual pseudoknot. We provide new, easily computable isotopy and homotopy invariants for classical and virtual pseudodiagrams. We also give tables of unknotting numbers for homotopically trivial pseudoknots and homotopy classes of homotopically nontrivial pseudoknots. Sinc…
Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using ``Morse-Bott'' techniques. As an applicat…
Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in n-dimensional Euclidean space, for n greater than or equal to 4. At the end of their paper they conjecture that when n is odd, the terms on the antidiagonal on the second page precisely give the space of primitive Feynman diagrams related to the theory of Vassiliev invariants. In this paper we prove that conjecture. This has the application that the path components of the terms of the Taylor tower for the space of classical long knots are in one-to-one correspondence with quot…
Every second flat Reidemeister move of knot projections can be decomposed into two types thorough an inverse or direct self-tangency modification, respectively called strong or weak, when orientations of the knot projections are arbitrarily provided. Further, we introduce the notions of strong and weak (1, 2) homotopies; we define that two knot projections are strongly (resp. weakly) (1, 2) homotopic if and only if two knot projections are related by a finite sequence of first and strong (resp. weak) second flat Reidemeister moves. This paper gives a new necessary and sufficient condition that…
A counterexample to a conjecture of Plotnick on knot exteriors
该研究针对Plotnick1983年关于四维纽结补空间同伦型的猜想，构造反例证明三叶结的两个分支扭转自旋外部为简单同伦等价，核心是构造满足特定条件的$\text{SL}_2(\text{F}_5)$整数群环元素。
We prove that independence complex of a bipartite circle graph is homotopy equivalent to a wedge of spheres, resolving a conjecture posed by Przytycki and Silvero. As a corollary, we obtain that extreme Khovanov spectrum, $\mathcal{X}_{j_{extreme}}$ is homotopy equivalent to a wedge of spheres. In particular, the extreme Khovanov homology has no torsion.
Generalized square knots and homotopy 4-spheres
本文通过分析nR-链环研究同伦4-球面，证明在特殊2-柄附着于广义方结时，无1-柄且仅有两个2-柄的同伦4-球面微分同胚于标准4-球面，并构造了广义性质R猜想的潜在反例族。
We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite type knot invariants, etc. Some open questions and conjectures of interest are given throughout.
A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first non-trivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence …
Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the ci…
A Weiss-Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots
本文建立了嵌入空间的伪同调结果，描述了块嵌入与普通嵌入在同伦度范围内的差异，并通过分析得出长结空间的同伦类型，包括同调群的计算。
Given a 3-manifold $Y$ and a free homotopy class in $[S^1,Y]$, we investigate the set of topological concordance classes of knots in $Y \times [0,1]$ representing the given homotopy class. The concordance group of knots in the 3-sphere acts on this set. We show in many cases that the action is not transitive, using two techniques. Our first technique uses Reidemeister torsion invariants, and the second uses linking numbers in covering spaces. In particular, we show using covering links that for the trivial homotopy class, and for any 3-manifold that is not the 3-sphere, the set of orbits is in…
On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant
本文计算了弧或圆嵌入空间的第一个非平凡同伦群，在单连通4流形情形回答Arone-Szymik问题，并在3维情形给出Vassiliev型普适纽结不变量。
Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topolog…
The homotopy cardinality of the representation category for a Legendrian knot
本文研究 Legendrian 纽结表示范畴，证明其 $n$ 维表示范畴的同伦基数是 $n$ 色 ruling polynomial 的倍数，并给出表示等价与共轭 DGA 同伦的关系及 Lagrangian concordance 应用。
Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to produce smooth circle and cyclic group actions on certain exotic spheres.
Strong and weak (1, 3) homotopies are equivalence relations on knot projections, defined by the first flat Reidemeister move and each of two different types of the third flat Reidemeister moves. In this paper, we introduce the cross chord number that is the minimal number of double points of chords of a chord diagram. Cross chord numbers induce a strong (1, 3) invariant. We show that Hanaki's trivializing number is a weak (1, 3) invariant. We give a complete classification of knot projections having trivializing number two up to the first flat Reidemeister moves using cross chord numbers and t…
Free circle actions on certain simply connected $7-$manifolds
研究确定哪些非负整数k,l和同调7球面Σ使kS²×S⁵#lS³×S⁴#Σ具有自由光滑圆作用
We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group of knots in the 3-sphere that ties in local knots. We prove that the trivial free homotopy class in every 3-manifold other than the 3-sphere contains an infinite family of knots, all topologically concordant, but not smoothly almost concordant to one another. Then, in every lens space and for every free homotopy class, we find a pair of topologically concord…
In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the ``original knot'' fibers over the circle then the wild knot $Λ$ also fibers over the circle. As a consequence, the universal covering of $\mathbb{S}^{3}-Λ$ is $\mathbb{R}^{3}$. We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold.
We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families of distinct homotopy-ribbon disks with homotopy equivalent exteriors, with potential relevance to the Slice-Ribbon Conjecture. We show that any fibered ribbon 2-knot can be obtained by doubling infinitely many different slice disks (sometimes in different contractible 4-manifold…
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。