All frame-spun knots are slice

Gluck twisting roll spun knots

We show that the smooth homotopy 4-sphere obtained by Gluck twisting the m-twist n-roll spin of any unknotting number one knot is diffeomorphic to the standard 4-sphere, for any pair of integers (m,n). It follows as a corollary that an infinite collection of twisted doubles of Gompf's infinite order corks are standard.

对旋结桥分解的Kirby-Thompson不变量的界

Bounding the Kirby-Thompson invariant of spun knots

本文基于旋结桥分解，首次给出旋结的Kirby-Thompson不变量的显著界，特别证明了旋结结的Kirby-Thompson不变量为15。

分支扭转旋纽的群

The groups of branched twist-spun knots

该研究根据3-流形群刻画经典纽结的分支扭转旋的群，给出基于$PD_3$-群的代数猜想刻画，并证明每个群对应有限个分支扭转旋。

关于扭转三叶结的纽结广群

On the knot quandle of the twist-spun trefoil

该研究探讨扭转三叶结的纽结广群性质，证明3-、4-、5-扭转三叶结的纽结广群分别与16-、24-、600-胞腔相关广群同构，且其基数有限当且仅当扭转次数1≤m≤5，并用正则镶嵌理论解释了该现象。

多项式纽结的2维球面

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。

Constructing a broken Lefschetz fibration of S^4 with a spun or twist-spun torus knot fiber

Much work has been done on the existence and uniqueness of broken Lefschetz fibrations such as those by Auroux et al., Gay and Kirby, Lekili, Akbulut and Karakurt, Baykur, and Williams, but there has been a lack of explicit examples. A theorem of Gay and Kirby suggests the existence of a broken Lefschetz fibration of S^4 over S^2 with a 2-knot fiber. In the case of a spun or twist-spun torus knot, we present a procedure to construct such fibrations explicitly. The fibrations constructed have no cusps nor Lefschetz singularities.

$\mathbb{R}^4$ 的标准立方化上旋转三叶结的最小面积

Minimal area of the spun trefoil knot on the canonical cubulation of $\mathbb{R}^4$

本文定义了$\mathbb{R}^4$标准立方化中立方2-结的最小面积不变量，并在弱最小情况下确定了旋转三叶结实现该结型所需的最小面积。

对旋转 2 - 纽结补空间进行三角剖分

Triangulating Spun 2-Knot Complements

研究如何对通过旋转1 - 纽结得到的2 - 纽结外部进行三角剖分，核心方法是提出算法并在Regina中实现，主要贡献是给出了由至多八个交叉的1 - 纽结得到的外部的三角剖分。

扭转-滚动 spun 结与翻转扭转环面的分支覆盖

Branched covers of twist-roll spun knots and turned twisted tori

本文证明了扭转-滚动 spun 结及其与平凡射影平面连通和的双分支覆盖在增加扭转后的光滑保持性，并据此确认了若干同伦 $\mathbb{CP}^2$ 和同伦 4-球分别微分同胚于标准流形。

High-codimensional knots spun about manifolds

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…

Braiding surface links which are coverings over the standard torus

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun $T^2$-knots and turned spun $T^2$-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an upper estimate of the braid index of a torus-covering link. In particular we show that the turned spun $T^2$-knot of the torus $(2,\,p)$-knot has the braid index four.

通过四维拓扑研究纽结与链环的交叉数

The crossing numbers of knots and links via 4D topology

该研究通过四维拓扑将链环交叉数与旋转带状环面纽结的弦指数关联，证明了链环连通和下交叉数的可加性。

Circle-valued Morse theory for frame spun knots and surface-links

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. …

Trisections and spun 4-manifolds

We study trisections of 4-manifolds obtained by spinning and twist-spinning 3-manifolds, and we show that, given a (suitable) Heegaard diagram for the 3-manifold, one can perform simple local modifications to obtain a trisection diagram for the 4-manifold. We also show that this local modification can be used to convert a (suitable) doubly-pointed Heegaard diagram for a 3-manifold/knot pair into a doubly-pointed trisection diagram for the 4-manifold/2-knot pair resulting from the twist-spinning operation. This technique offers a rich list of new manifolds that admit trisection diagrams that ar…

0-Shake Slice Knots are Slice

A proof that $0-$shake slice knots are slice.

On Legendrian products and twist spuns

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In order to do this, we prove a few structural results on the bilinearised Legendrian contact homology and augmentation variety of a twist spun. In addition, we show that the threefold Bohr-Sommerfeld covers of the Clifford torus and Chekanov torus are not twist spuns.

用于 Gluck 扭转的无穷多个标准三分图

Infinitely many standard trisection diagrams for Gluck twisting

针对 Gay 与 Meier 关于 Gluck 扭转三分图标准性的问题，本文证明拧成 $(p+1,p)$-环面纽结在 $p\ge2$ 时所得三分图均为标准图。

Polynomial splittings of correction terms and doubly slice knots

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth-Szabó correction terms as smooth double sliceness obstructions vanish for both knots. Recently, Jeffrey Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. As an application, we give a new example of such knots that is distinct from Meier's knots modulo doubly slice knots.

The Conway knot is not slice

A knot is said to be slice if it bounds a smooth properly embedded disk in the 4-ball. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

Topologically slice $(1,1)$-knots which are not smoothly slice

We prove that there are infinitely many $(1,1)$-knots which are topologically slice, but not smoothly slice, which was a conjecture proposed by Béla András Rácz.

D不变量对一类代数可切片纽结的可切片性的阻碍

D invariants obstruction to sliceness of a class of algebraically slice knots

本文利用二重分支覆盖的d不变量，证明了一组代数可切片纽结实际上是不可切片的，从而给出了可切片性的新阻碍。

Topologically slice knots that are not smoothly slice in any definite 4-manifold

We prove that there exist infinitely many topologically slice knots which cannot bound a smooth null-homologous disk in any definite 4-manifold. Furthermore, we show that we can take such knots so that they are linearly independent in the the knot concordance group.

Knot Floer Filtration Classes of Topologically Slice Knots

The knot Floer complex and the concordance invariant $\varepsilon$ can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to $\mathbb{N} \times \mathbb{N}$ and consists of topologically slice knots.

On rational sliceness of Miyazaki's fibered, $-$amphicheiral knots

We prove that certain fibered, $-$amphicheiral knots are rationally slice. Moreover, we show that the concordance invariants $ν^+$ and $Υ(t)$ from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots.

用不可光滑逼近的拓扑切片圆盘光滑切片纽结

Smoothly slice knots with smoothly non-approximable topological slice discs

本文构造出无穷多个光滑切片纽结，其拓扑切片圆盘无法被光滑切片圆盘逼近，解决了光滑与拓扑切片圆盘的逼近问题。

Equivariant topological slice disks and negative amphichiral knots

We show that any strongly negative amphichiral knot with a trivial Alexander polynomial is equivariantly topologically slice.

2-纽结的切片深度

Slice depth of 2-knots

本文提出2-纽结的切片深度概念，并针对若干特定经典纽结类别的n-扭转自旋给出切片深度上界。

A characterization of the Murasugi polynomial of an equivariant slice knot

We characterize the Murasugi polynomial of an equivariant slice knot by proving a conjecture of J. Davis and S. Naik.

A note on the topological sliceness of some 2-bridge knots

We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.

Linear independence of rationally slice knots

A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two.

A note on cabled slice knots and reducible surgeries

We consider the question of when a slice knot admits a reducible Dehn surgery. By analyzing the correction terms associated to such a surgery, we show that slice knots cannot admit surgeries with more than two summands. We also give a necessary Heegaard Floer theoretic condition for a positive cable of a knot to be slice.

Doubly slice knots and metabelian obstructions

For $\ell >1$, we develop $L^{(2)}$-signature obstructions for $(4\ell-3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell>1$, we construct an infinite family of knots on which our obstructions are non-zero, but for which double sliceness is not obstructed by any previously known invariant.

拓扑切片与有理切片纽结

Topologically and rationally slice knots

本文证明拓扑切片且有理切片纽结的光滑协边群含有 $\mathbb{Z}^\infty$ 子群，从而首次给出无穷多个非二阶的同时拓扑切片与有理切片纽结，并推出存在无穷多个强有理切片但非切片的拓扑切片纽结。

Distinguishing topologically and smoothly doubly slice knots

We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show that the subgroup of the double concordance group consisting of smoothly slice, topologically doubly slice knots is infinitely generated. As a corollary, we produce an infinite collection of rational homology 3-spheres that embed in $S^4$ topologically, but not smoothly.

hyperbolic fibered slice knots with right-veering monodromy

We construct a hyperbolic fibered slice knot with right-veering monodromy, giving a negative answer to the question posed by Hubbard-Kawamuro-Kose-Martin-Plamenevskaya-Raoux-Truong-Turner.

On the Slice-Ribbon Conjecture for Montesinos knots

We establish the slice-ribbon conjecture for a large family of Montesinos' knots by means of Donaldson's theorem on the intersection forms of definite 4-manifolds.

Doubly slice odd pretzel knots

We prove that an odd pretzel knot is doubly slice if it has $2n+1$ twist parameters consisting of $n+1$ copies of $a$ and $n$ copies of $-a$ for some odd integer $a$. Combined with the work of Issa and McCoy, it follows that these are the only doubly slice odd pretzel knots.

Non-slice linear combinations of algebraic knots

We show that the subgroup of the knot concordance group generated by links of isolated complex singularities intersects the subgroup of algebraically slice knots in an infinite rank subgroup.

Distinguishing slice disks using knot Floer homology

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a slice disk of a composite knot, we define a numerical stable diffeomorphism invariant called the rank. This can be used to show that a slice disk is not a boundary connected sum, and to give lower bounds on the complexity of certain hyperplane sections of the slice disk.

A construction of slice knots via annulus modifications

We define an operation on homology ${B}^4$ which we call an $n$-twist annulus modification. We give a new construction of smoothly slice knots and exotically slice knots via $n$-twist annulus modifications. As an application, we present a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. Also, we relate $n$-twist annulus modifications to $n$-fold annulus twists which was first introduced by Osoinach, then has been used by Abe and Tange to construct smoothly slice knots. Furthermore we consider $n$-twist annulus modificati…

Order 2 Algebraically Slice Knots

The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebraically slice knots also contain elements of finite order; in fact it contains an infinite subgroup generated by elements of order 2.

切片纽结与纽结协和

Slice knots and knot concordance

本文是Winterbraids XI讲座讲稿，系统介绍切片纽结、协和等价关系、其与$\mathbb{R}^4$奇异光滑结构的联系，以及协和群滤过和卫星操作。

Doubly slice knots with low crossing number

A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2,977 prime knots in that range. The techniques involved in this analysis include considerations of the Alexander module and signature functions as well as calculations of the twisted Alexander polynomials for higher-order branched covers. We give explicit illustrations of the dou…

Pseudo-slice knots

For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that if the metabolizer has a basis represented by a strongly slice link then K is slice. The question has been asked as to whether it is sufficient that each basis element is represented by a slice knot to assure that K is slice. For genus one knots this is of course true; here we present a genus two counterexample.

具有Alexander多项式1和高解结数的光滑切片纽结

Smoothly slice knots with Alexander polynomial 1 and high unknotting number

本文证明存在一个同时具有光滑双切片性、双向手性、Alexander多项式为1且解结数为5的纽结，揭示了这些拓扑性质可以共存。

A construction of slice knots via annulus twists

We give a new construction of slice knots via annulus twists. The simplest slice knots obtained by our method are those constructed by Omae. In this paper, we introduce a sufficient condition for given slice knots to be ribbon, and prove that all Omae's knots are ribbon.

Virtual knot cobordism and bounding the slice genus

In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an invariant of virtual knot concordance. The graded genus is remarkably effective as a slice obstruction, and we develop an algorithm that applies virtual unknotting operations to determine the slice genus of many virtual knots with 6 or fewer crossings.

Rasmussen invariant, Slice-Bennequin inequality, and sliceness of knots

We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and, to the best of our knowledge, were previously unknown.

局部结、ν⁺-尖结和有理切 genus

Local Knots, $ν^+$-Sharp Knots, and Rational Slice Genus

本文研究了局部结和ν⁺-尖结的性质，证明了这些结在有理同调3-球体中切 genus 等于原结的切 genus，利用Heegaard Floer不变量ν⁺得出相关结论。

Knot reversal acts non-trivially on the concordance group of topologically slice knots

We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if T denotes the concordance group of topologically slice knots and R is the involution of T induced by string reversal, then T/Fix(R) contains an infinitely generated free subgroup. The result remains true modulo the subgroup of T generated by knots with trivial Alexander polynomial.

A note on rationally slice knots

Kawauchi proved that every strongly negative amphichiral knot $K \subset S^3$ bounds a smoothly embedded disk in some rational homology ball $V_K$, whose construction a priori depends on $K$. We show that $V_K$ is independent of $K$ up to diffeomorphism. Thus, a single 4-manifold, along with connected sums thereof, accounts for all known examples of knots that are rationally slice but not slice.

Counterexamples to Kauffman's Conjectures on Slice Knots

In 1982 Louis Kauffman conjectured that if a knot in the 3-sphere is a slice knot then on any Seifert surface for that knot there exists a homologically essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explictly expressed in terms of invariants of such curves on the Seifert surface. We give counterexamples to Kauffman's conjecture, that is, we exhibit (smoo…

Algebraic and Heegaard-Floer invariants of knots with slice Bing doubles

If the Bing double of a knot K is slice, then K is algebraically slice. In addition, Heegaard--Floer concordance invariants developed by Ozsvath-Szabo and by Manolescu-Owens vanish on K.

Rasmussen s-invariants of satellites do not detect slice knots

We present a large family of knots for which the Rasmussen s-invariants of arbitrary satellites do not detect sliceness. This answers a question of Hedden. The proof hinges on work of Kronheimer-Mrowka and Cochran-Harvey-Horn.