Wenzhao Chen
In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category.
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We study the double slice genus of a knot, a natural generalization of slice genus. We define a notion called band number, a natural generalization of band unknotting number, and prove it is an upper bound on double slice genus. Our bound is based on an analysis of broken surface diagrams and embedding properties of 3-manifolds in 4-manifolds.
A slice genus lower bound from sl(n) Khovanov-Rozansky homology
我们证明，扰动sl（n）Khovanov-Rozansky链接同调的定义给出了结的切片亏格的下界。作为一个推论，这产生另一个证明米尔诺猜想片属的环面结。
Equivariant Double-Slice Genus, Stabilization, and Equivariant Stabilization
本文研究了强可逆结的等变双切环面和超切环面，通过下界证明发现了一类具有特定性质的结，并探讨了其在稳定化距离中的应用。
The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the classical signature function of the knot to give a new lower bound for the doubly slice genus. We combine this with an upper bound due to C. McDonald to prove that for every nonnegative integer $N$ there is a knot where the difference between the slice and doubly slice genus is exactly $N$, refining a result of W. Chen which says this difference can be arbitrarily large.
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.
This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $\mathbb{Z}$-slice genus. As an application, we show that the $(n,1)$-cable of any 3-genus 1 knot (e.g. the figure 8 or trefoil knot) has topological slice genus at most 1. Further, we show that the lower bounds on the slice genus coming from the Tristram-Levine and Casson-Gordon signatures canno…
On the concordance genus of topologically slice knots
纽结K的协调亏格是光滑协调于K的所有纽结的最小Seifert亏格。协调亏格的下界是4球亏格，上界是塞弗特亏格。我们给出了K的协调亏格的一个下界，它来自于K的纽结Floer复形.作为应用，我们证明了存在4球亏格等于1且任意大的协调亏格的拓扑切片纽结。
We construct an infinite family of topologically slice 2--component boundary links $\ell_i$, none of which is smoothly concordant to a split link, such that $g_4(\ell_i)=i$.
On the slice genus of links
定义了链环的Casson-Gordon σ-不变量，并给出了链环的切片亏格的一个下界.作为一个例子，我们研究了一个家庭的两个组成部分的链接属h，并表明，他们的切片属是h，而Murasugi-Tristram不等式并不妨碍这个链接从边界的4球环。
We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and Manolescu give both a slice genus and a clasp number bound.
We introduce a new class of links for which we give a lower bound for the slice genus $g_*$, using the generalized Rasmussen invariant. We show that this bound, in some cases, allows one to compute $g_*$ exactly; in particular, we compute $g_*$ for torus links. We also study another link invariant: the strong slice genus $g_*^*$. Studying the behaviour of a specific type of cobordisms in Lee homology, a lower bound for $g_*^*$ is also given.
An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is …
Slice genus, $T$-genus and $4$-dimensional clasp number
本文推广了切片结的$T$-亏格刻画，为切片亏格提供了$3$维刻画，证明$T$-亏格控制$4$维正交叉数，引入带状$T$-亏格并探讨其与切片结、带状结及着色链环的关联。
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples of knots where this determines the topological slice 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。
We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure eight knot is not smoothly slice, as shown by Dai--Kang--Mallick--Park--Stoffregen in 2022. The main technical input of our argument consists of gauge-theoretic obstructions to smooth small-genus surfaces representing certain homology classes in $\mathbb{CP}^2\#\mathbb{CP}^2$ proved by Bryan in the 1990s.
In this paper, we study reducible surgeries on knots in $S^3$. We develop thickness bounds for L-space knots that admit reducible surgeries, and lower bounds on the slice genus for general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots, which was mostly worked out in \cite{DeY21b}. We also provide a new upper bound on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots. Our techniques involve the $d$-invariants and mapping cone formula from H…
We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebr…
Higher-order Analogues of the Slice Genus of a Knot
对于某些类别的结，我们定义几何不变量称为高阶属。这些不变量中的每一个都是纽结的切片亏格的细化。我们发现下界的高阶属在某些冯诺依曼$ρ$ -不变量，我们称之为高阶签名。高阶属提供了一个完善的Grope过滤的结一致性组。
By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower bound to the non-orientable slice genus $γ_4(K)$ in terms of the signature $σ(K)$ and the concordance invariants $V_i(\overline{K})$, which strengthens a previous bound given by Batson, and which coincides with Ozsváth-Stipsicz-Szabó's bound in terms of their $\upsilon$ invariant for L-space knots and quasi-alternating knots. A curious feature of our bound is superadditivity, implying, for instance, that the bound on the stable non-orientable genus is sometimes better than the one on $γ_4(K)$.
On surgery curves for genus one slice knots
若纽结K是三维球面中亏格为1的Seifert曲面F的边界，且F包含诱导框架为0的本质简单闭曲线α且是光滑切片的，则K是光滑切片的.从推测上讲，相反的情况也成立。已知如果K是切片，则在这样的alpha的代数和谐类上存在强约束，并且认为这些约束可能暗示alpha至少是代数切片。我们提出了一个反例，在这个过程中，我们回答否定的问题库珀和相关的结果考夫曼的问题。本文的结果依赖于K的Casson-Gordon不变量和α的代数不变式之间的相互作用。
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, consequently, for bounding the topological slice genus above. As applications we give new upper bounds on the algebraic genera of torus knots and satellite knots.
A note on realizing triples of slice, concordance and Seifert genera
该研究证实Kearney的猜想，即满足1≤a≤b≤c的任意整数三元组(a,b,c)都可作为纽结的光滑切片亏格、光滑配边亏格与Seifert亏格实现。
We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertible slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum $\#^n K$ is at least $n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.
The twisting number of a ribbon knot is bounded below by its doubly slice genus
该研究证明带状纽结的扭转数以双重切片亏格为下界，构造出扭转数任意高的带状纽结解决Tanaka的问题，并确定了部分12交叉纽结的未知双重切片亏格。
On definite lattices bounded by integer surgeries along knots with slice genus at most 2
本文分类了以右手三叶结及切片亏格至多2的纽结整数手术为边界的正定交形式，借助Yang--Mills瞬子规范理论与Heegaard Floer修正项完成证明。
Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by $\mathcal{F}_n$. It has been shown that $\mathcal{F}_n/\mathcal{F}_{n.5}$ is a very large group for $n\ge 0$. For a generalization to the setting of links the third author showed that $\mathcal{F}_{n.5}/\mathcal{F}_{n+1}$ is non-trivial. In this paper we provide evidence that for knots $\mathcal{F}_{0.5}=\mathcal{F}_1$. In particular we prove that every genus 1 algebraically slice knot is 1-solvable.
On calculating the slice genera of 11- and 12-crossing knots
本文通过计算机搜索亏格一协边、代数亏格、Seifert形式新阻碍及Donaldson对角化定理，确定了绝大多数11和12交叉纽结的切片亏格，仅剩少数情况未知。
A virtual knot is an equivalence class of embeddings of $ S^1 $ into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of $ S^1 $ into a thickened surface $ Σ_g \times I $; what is the minimal genus of oriented surfaces $ S \hookrightarrow M \times I $ with the embedded $ S^1 $ as boundary, where $ M $ is an oriented $ …
Khovanov homology and the slice genus
我们使用李的Khovanov同调的工作来定义一个结不变量。我们证明了s（K）是一个协调不变量，它为K的切片亏格提供了一个下界.作为推论，我们给出了Milnor猜想的一个纯组合证明。
Topological minimal genus and $L^2$-signatures
利用vonNeumann-Cheeger-Gromov $ρ$ -不变量，得到了拓扑四维带边流形中表示二维同调类的局部平坦曲面的最小亏格的新下界.作为一个应用，我们的结果被用来研究切片亏格的纽结。我们举例说明任意大的切片属，我们的下界是最佳的，但所有以前已知的不变量消失。
We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.
For a knot $K$, the doubly slice genus $g_{ds}(K)$ is the minimal $g$ such that $K$ divides a closed, orientable, and unknotted surface of genus $g$ embedded in $S^4$. In this paper, we identify the doubly slice genera of 2909 of the 2977 prime knots which have a crossing number of 12 or fewer.
The Concordance Genus of 11--Crossing Knots
纽结的协调亏格是其协调类中任何纽结的最小亏格。它的上界是由纽结的亏格所限定，下界是由切片亏格所限定，这是两个研究得很好的不变量。本文研究了11-交叉素纽结的协调亏格。这种分析解决了552个素数11-交叉结中的533个的一致性属。附录中的文件给出了协调图59节被发现是一致的低属的节，包括零协调的30个11-交叉节已知的切片。
On the slice genus of generalized algebraic knots
本文通过构造代数切片但四亏格为二的代数纽结线性组合，推广了Hedden–Kirk–Livingston的迭代环面纽结例子，并利用Casson–Gordon不变量与缆绳公式给出亏格下界。
We introduce the notion of ascent sliceness of virtual knots. A representative of a virtual knot is an embedding $ S^1 \hookrightarrow Σ_{g} \times I $, for $ Σ_g $ a closed connected oriented surface of genus $ g $; the virtual knot represented is slice if there exists a pair consisting of a disc $ D $ and an oriented $ 3 $-manifold $ M $, such that $ D \hookrightarrow M \times I $, $ \partial M = Σ_{g} $, and $ \partial D = S^1 $ (the image of the embedding). This definition of sliceness exemplifies that a cobordism of virtual links is a pair consisting of a surface and a $ 3 $-manifold; in …
Pseudo-slice knots
当n >1时，如果S^{2n+1}中的2n-1纽结球K的塞弗特形式有代谢子，则纽结是切片的。卡森和戈登证明了这在3维（n = 1）中是错误的。然而，在三维情况下，如果代谢者具有由强切片链接表示的基础，则K是切片。这个问题已经被问到，是否它是足够的，每个基本元素表示的切片结，以确保K是切片。对于亏格1的纽结，这当然是正确的;这里我们提出一个亏格2的反例。
A 2-component oriented link in $S^3$ is called weakly doubly slice if it is a cross-section of an unknotted sphere in $S^4$, and strongly doubly slice if it is a cross-section of a 2-component trivial spherical link in $S^4$. We give the first example of 2-component boundary links which are weakly doubly slice but not strongly doubly slice. We also introduce a new invariant $g_{st}$ of homotopically trivial links that measures the failure of a link from being strongly doubly slice and that bounds the doubly slice genus $g_{ds}$ from below. Our examples have arbitrarily large doubly slice genus…
A 4-dimensional rational genus bound
本文引入纽结的有理切片亏格，用Heegaard Floer τ不变量给出下界，并应用于Floer简单纽结及产生切片亏格无界序列等。
Slice genus bound in $DTS^2$ from $s$-invariant
本文证明了 Manolescu-Willis 关于 $\mathbb{RP}^3$ 中纽结的 $s$-不变量给出 $TS^2$ 单位圆盘丛中零同调切片亏格下界的猜想，并推广到非零同调情形，在特殊情形下给出证明。
A geometric proof of the Milnor conjecture
给出了正纽结的Seifert亏格等于其切片亏格的几何证明。一个组合不变量，给出了一个下界的切片属，制定了任意的结。讨论了该不变量的性质和应用。
On slicing invariants of knots
结的切片数$u_s(K)$是将$K$转换为切片结所需的最小交叉更改数。该不变量的上界为解结数，下界为切片亏格$g_s(K)$。我们表明，对于许多结，以前的界限unknotting数得到的Ozsvath和Szabo和作者实际上给界的切片数。利文斯顿定义了另一个不变量$U_s(K)$，它考虑了交叉点的符号变化，以得到一个切片结，它的上限是切片数，下限是切片亏格。我们展示了一个无限族的节$K_n$与切片亏格$n$和Livingston不变量大于$n$。我们的界限是基于限制（使用唐纳森的对角化定理或Heegaard弗洛尔同调）的交叉形式的四流形有界的双分支覆盖的一个结。
Unknotting sequences for torus knots
一个纽结的解结数由它的切片亏格从下而上限定。众所周知，环面纽结的属数和解纽结数是一致的。本文刻画了亏格界为尖的拟正纽结：一个拟正纽结的切片亏格等于它的解纽结数，当且仅当给定的纽结出现在环面纽结的解纽结序列中.
A derivative of an algebraically slice knot $K$ is an oriented link disjointly embedded in a Seifert surface of $K$ such that its homology class forms a basis for a metabolizer $H$ of $K$. We show that for a genus three algebraically slice knot $K$, the set $\{ \barμ_{\{γ_1,γ_2,γ_3\}}(123) - \barμ_{\{γ'_1,γ'_2,γ'_3\}}(123)| \{γ_1,γ_2,γ_3\}$ and $\{γ'_1,γ'_2,γ'_3\}$ are derivatives of $K$ associated with a metabolizer $H\}$ contains $n\cdot \mathbb{Z}$ where $n$ is determined by a Seifert form of $K$ and a metabolizer $H$. As a corollary, we show that it is possible to realize any integer as a …
Slice knots which bound punctured Klein bottles
研究了S^3中约束克莱因瓶的纽结的性质，使得纽结的推射与纽结的环绕数为零，即具有零框架。这是由文献中关于亏格为1的切片结点的许多结果所激发的，例如，对于代数切片结点，在亏格为1的Seifert曲面上存在同调本质零自连接的简单闭曲线.给定一个结K，它是一个具有零框架的克莱因瓶F的边界，我们证明了F的任何盘带形式中的取向保持带的核心J具有零自链接。我们证明了这样一个K是Z[1/2]-同调4-球中的切片当且仅当J也是，一个比目前已知的亏格切片结更强的结果。作为应用，我们证明了给定纽结K和J以及任意奇数p，K和J的（2，p）索是Z[1/2]-协调的当且仅当K和J是Z[1/2]-协调的.特别地，如果一个纽结K的（2，1）-索是切片，则K在Z[1/2]-同调球中是切片。
In the 60's Levine proved that if $R$ is a slice knot, then on any genus $g$ Seifert surface for $R$ there is a $g$ component link $J$, called a derivative of $R$, on which the Seifert form vanishes. Many subsequent obstructions to $R$ being slice are given in terms of slice obstructions of $J$. Many of these obstructions can be derived from a 4-manifold called a null-bordism. Recently the authors proved that that it is possible for $R$ to be slice without $J$ being slice, disproving a conjecture of Kauffmann from the 80's. In this paper we cut open these null-bordisms in order to derive new o…
Algebraic and Heegaard-Floer invariants of knots with slice Bing doubles
如果一个纽结K的Bing双曲是切片的，那么K是代数切片的。此外，Ozsvath-Szabo和Manolescu-Owens发展的Heegaard-Floer和谐不变量在K上消失了.
Computations of the slice genus and the unknotting number of links via machine learning
本文利用强化学习和贝叶斯优化为链环的切片亏格、解纽结数及强切片亏格计算新的上界，结合已知下界获得精确值，并重现了解纽结数的非可加性反例。
On the slice genus of quasipositive knots in indefinite 4-manifolds
本文针对满足特定 Seiberg--Witten 不变量条件的不定4-流形，证明了以准正纽结为边界的嵌入曲面的亏格新下界，并在零同调情形下精确确定最小亏格等于切片亏格。
We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.
Khovanov homology and equivariant surfaces
本文通过精细化 Bar-Natan 同调构造新数值不变量，证明强可逆纽结的等变切片亏格与同痕等变切片亏格之差可任意大。
We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabolic and hyperbolic linking forms. We apply the double $L$-groups in high-dimensional knot theory to define an invariant for doubly-slice $n$-knots. We prove that the "stably doubly-slice implies doubly-slice" property holds (algebraically) for Blanchfield forms, Seifert forms and…
Whitehead Doubles and Non-Orientable Surfaces
本文研究 Figure 8 纽结 Whitehead double 的切片性，综述相关阻碍技术，并改进 $t$-扭转 Whitehead doubles 的不可定向 4 维亏格界，构造到缆线纽结的亏格 1 配边。
A higher-order genus invariant and knot Floer homology
众所周知，纽结弗洛尔同调检测纽结的亏格和亚历山大多项式。我们调查是否结弗洛尔同调的$K$检测到更多的结构的最小亏格塞弗特表面的$K$。我们定义了一个代数切片，亏格为1的纽结的不变量，并举例说明纽结的弗洛尔同调不能检测到这个不变量。最后，我们注意到某些元阿贝尔$L^2$ -签名从下面绑定这个不变量。
Relative genus bounds in indefinite four-manifolds
本文研究不定闭四维流形中带边界曲面的相对亏格，利用伴随不等式与 $10/8+4$ 定理给出界，并以 H-slice 障碍检测异乎寻常光滑结构。
We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden-Livingston-Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.
Equivariant knots and knot Floer homology
本文使用纽结Floer同调定义了等变同痕不变量，给出了等变切片亏格下界并构造了强可逆切片纽结族，同时应用于检测切片圆盘奇异对及扩展稳定化距离结果。
For each $g>0$ we give infinitely many knots that are strongly negative amphichiral, hence rationally slice and representing 2-torsion in the smooth concordance group, yet which do not bound any locally flatly embedded surface in the 4-ball with genus less than or equal to $g$. Our examples also allow us to answer a question about the 4-dimensional clasp number of knots.
On the values taken by slice torus invariants
该研究聚焦切片环面不变量空间，以稳定光滑切片亏格刻画其在给定纽结上的取值集合，发现局部Thom猜想的证明可直接推导切片环面不变量的存在性，无需依赖纽结同调理论的显式构造。
Covering link calculus and iterated Bing doubles
本文给出了纽结的迭代Bing双链是条链的一个新的几何障碍：当n>1时，纽结的第（n+1）次迭代Bing双链是有理条链当且仅当纽结的第n次迭代Bing双链是有理条链.证明的主要技术是一个覆盖链接结构简化给定的链接。我们证明了某些类似的几何障碍，以及n <= 1。我们的结果是尖锐的，足以得出这样的结论，当结合代数不变式，如果第n次迭代的Bing双结是切片的一些n，那么结是代数切片。此外，我们的几何参数适用于光滑的情况下，显示Ozsvath-Szabo和Manolescu-Owens不变量给迭代Bing双打切片的障碍。这些结果推广了Harvey，Teichner，Cimasoni，Cha和Cha-Livingston-Ruberman最近的结果.作为另一个应用，我们给出了明确的例子，代数切片结与非切片迭代Bing双考虑冯诺依曼rho不变和合理的结和谐。这些例子的改进版本，考虑到Cochran-Orr-Teichner过滤。
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