Sebastian Baader, Pierre Dehornoy, Livio Liechti
We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
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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。
Invertible knot concordances and prime knots
Kirby和Lickorish证明了3-球面中的每个纽结都与一个素纽结和谐，等价地说，每个和谐类都包含一个素纽结。我们在这里证明了他们的结果可以得到加强：3-球面中的每一个纽结都可逆地与一个素纽结和谐。一个推论是每个双和谐类都包含一个素结。
We study concordance of virtual knots. Our main result is that a classical knot K is virtually slice if and only if it is classically slice. From this we deduce that the concordance group of classical knots embeds into the concordance group of long virtual knots.
We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.
The knot Floer complex and the smooth concordance group
我们定义了一个新的光滑协调同态的基础上结Floer复杂和相关的协调不变量，。作为应用，我们证明了在光滑协调群中拓扑切片纽结的无限族是独立的。
If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.
We study 3-braid knots of finite smooth concordance order. A corollary of our main result is that a chiral 3-braid knot of finite concordance order is ribbon.
There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.
In this note we show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon. The proof makes use of representation varieties of the knot groups to $SO(N)$ and relations between them induced by a ribbon concordance.
On knots with infinite smooth concordance order
我们使用由格雷斯比、鲁伯曼和斯特尔定义的希格·弗洛尔障碍物来证明，在67节中，有46节通过11个交叉点，其和谐顺序以前是未知的，它们具有无限的和谐顺序。
We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated $(\text{odd},1)$-cables have infinite order in the concordance group and, among them, infinitely many are linearly independent. Furthermore, by taking $(2,1)$-cables of the aforementioned knots, we present an infinite family of knots which are strongly rationally slice but not slice.
Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invariants. Infection is a generalization of satellite operations, which has been previously studies. An infection by a string link can be thought of as grabbing a knot at multiple locations and then tying in a link. Cochran, Friedl and Techner showed that any algebraically slice knot is…
We review some recent results in knot concordance and homology cobordism. The proofs rely on various forms of Heegaard Floer homology. We also discuss related open problems.
The concordance genus of a knot, II
一个纽结K的协调亏格是所有与K协调的纽结中最小的三个亏格。对于总理节10个或更少的交叉有三个节的一致属是未知的。这三起案件现已解决。其中两种情况是解决使用不变量的莱文的代数和谐组。最后一种情况取决于扭曲的亚历山大多项式的使用，被视为卡森-戈登不变量。
In this paper we investigate the 0-concordance classes of 2-knots in $S^4$, an equivalence relation that is related to understanding smooth structures on 4-manifolds. Using Rochlin's invariant, and invariants arising from Heegaard-Floer homology, we will prove that there are infinitely many 0-concordance classes of 2-knots.
Turaev genus, knot signature, and the knot homology concordance invariants
我们给出了纽结签名、Ozsvath-Szabo τ不变量和Rasmussen不变量在纽结的Turaev亏格下的界。
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 prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.
A note on the concordance of fibered knots
要么支持紧切触结构的纤维结在光滑协调类中是唯一的，要么存在切片-带状猜想的纤维反例。
A survey of classical knot concordance
这是关于经典纽结和谐组的调查，为即将出版的纽结理论手册做准备。主题包括：和谐的基本定义;理论的代数和谐发展的莱文;理论的卡森-戈登不变量;应用拓扑手术发展的弗里德曼;顺利的结果的基础上微分几何技术，如唐纳森;过滤的和谐组发展的科克伦，奥尔和泰希纳;和关系到经典的三维纽结理论。还有一个简短的问题清单。
We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. We also generalize a theorem of Gabai about the super-additivity of the Seifert genus under band connected sum. Our result gives evidence for a conjecture of Gordon that ribbon concordance is a partial order on the set of knots.
Concordance Crosscap Number of a Knot
我们定义一个纽结的协调交叉数为与该纽结协调的所有纽结中的最小交叉数。四维crosscap数是光滑嵌入四维球中的非可定向曲面的最小第一贝蒂数，以结点为边界。显然，4维交叉覆盖数小于或等于一致交叉覆盖数。我们构造两个无穷序列的结来解释两者之间的差距。特别地，结$7_4$是示例之一。
It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.
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 show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger-Gromov-von Neumann rho-invariants for amenable groups developed by Cha and Orr and polynomial splittings of metabelian rho-invariants.
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。
Concordance invariants of knots are derived from the instanton homology groups with local coefficients, as introduced in earlier work of the authors. These concordance invariants include a 1-parameter family of homomorphisms $f_{r}$, from the knot concordance group to the reals. Prima facie, these concordance invariants have the potential to provide independent bounds on the genus and number of double points for immersed surfaces with boundary a given knot.
We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.
We present the complete classification of the subgroup of the classical knot concordance group generated by knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine crossing knots.
Obstructing 4-torsion in the classical knot concordance group
本文证明了：如果3-球面中纽结K的2-重分支覆盖的第一同调的阶由pm给出，其中p是3 mod 4的素同余且gcd（p，m）=1，则K在纽结和谐群中是无限阶的.这为4阶的经典纽结提供了障碍。特别是，有11个素数结与10个或更少的交叉是阶4的代数和谐组;所有的和谐是无限阶。另一个推论是，任何亚历山大多项式为5t^2 - 11 t + 5的纽结都是和谐的无穷阶;莱文证明了在高维中所有这样的纽结都是4阶的。
Knot Concordance and Torsion
设K是具有2重分支覆盖空间M的3-球面中的纽结。若对某个同余于3 mod 4的素数p，M的第一同调中的p-挠是奇指数循环的，则K在纽结和谐群中是无限阶的.作为一个应用，回想一下，一个任意纽结的n-扭曲的双重在Levine的代数和谐群中有4阶当且仅当n是正的，并且一些素同余到3 mod 4在4 n +1中有奇指数;我们证明了所有这样的纽结在纽结和谐群中都是无限阶的。作为第二个应用，如果某个3 mod 4的素同余在r中有奇指数，则2-桥纽结K（r，s）在纽结和谐群中有无穷阶。
Stabilizing Four-Torsion in Classical Knot Concordance
设$M_K$是纽结的2重分支覆盖$K in $ S^3 $. If $ H_1（M_K）= { \bf Z}_3 \oplus { \bf Z}_{3^{2 i}} \oplus G $ where 3 does not divide the order of $ G $ then $ K$在协调群中不是4阶的.这个障碍检测无限新的家庭的节点，代表元素的顺序4的代数和谐组，而不是顺序4的和谐。
Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group $\mathscr{VC}$. It is shown that for every concordance class $[K] \in \mathscr{VC}$, there is a $J \in [K]$ that is not band-pass equivalent to $K$ and an $L \in [K]$ that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that $v_{2,1}+v_{2,2} \pmod 2$ is a band-pass invariant but…
Topologically slice knots of smooth concordance order two
拓扑切片纽结的存在是无穷阶的纽结协调组遵循弗里德曼的工作拓扑手术和唐纳森的规范理论方法4流形。在这里，作为Ozsvath和Szabo的Heegaard-Floer理论的应用，我们证明了存在一个无穷子群的光滑和谐组生成的拓扑切片结的和谐阶2。此外，在这个子群中，没有非平凡元素可以用一个结来表示，该结的Alexander多项式为1。
In this paper we construct a sequence of integer-valued concordance invariants $ν_n(K)$ that generalize the Ozsváth-Szabó $ν$-invariant and the Hom-Wu $ν^+$-invariant.
We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming from knot Floer homology, and discuss applications to topologically slice knots, concordance genus, and concordance unknotting number.
The effect of infecting curves on knot concordance
使用卡森-戈登不变量、高阶亚历山大多项式以及冯-诺依曼rho不变量发现了结一致性的各种障碍。已经使用（迭代）加倍运算K=R（c，J）产生了示例，并且将这些视为由基结J和加倍运算符R的不变量参数化。本文介绍了一种新的协调性屏蔽方法。我们证明，通过改变S^3-R中的感染曲线c，可以构造出无穷多个不同的和谐类。不同的和谐类被发现，即使在固定的基础结，加倍运营商，并在亚历山大模块c的顺序。
The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, and concordance genus.
Ribbon concordance and the minimality of tight fibered knots
本文证明了所有紧纤维化纽结在带状合痕构成的偏序集中具有极小性，并给出了除8_15外至多8交叉的素极小纽结表。
We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot.
Knot mutation: 4-genus of knots and algebraic concordance
Kearton观察到突变可以改变结的一致性类别。仔细研究他的例子，发现它是4-属1，并有一个4-属0的突变体。本文的第一个目标是构造例子来证明对于任何一对非负整数m和n，存在一个4亏格m的纽结和一个4亏格n的突变体。本文的第二个结果是一个交叉变化公式的代数和谐类的纽结，然后应用它来证明的代数和谐类下突变的不变性。本文最后应用交叉变换公式给出了强正双曲纽结是代数切片的Long定理的一个简短的新证明。
Stable concordance of knots in 3-manifolds
本文将交不变量应用于奇异协调，研究了三维流形中的纽结和链环。由此产生的链接不变量推广的Arf不变量，模2 Sato-Levine不变量，和Milnor的三重环绕数。除了适合到惠特尼塔的一般理论，这些不变量提供了障碍，存在一个奇异的和谐，可以同伦稳定后的嵌入连接和$S^2\times S^2$。结果包括可定向三维流形中稳定切片链的分类，可定向曲面与圆的乘积的稳定纽结协调性，以及可定向三维流形中零同伦纽结的多个链的稳定链协调性.
On the knot Floer filtration of the concordance group
纽结Floer复形与相关的和谐不变量可用于定义平滑和谐群上的过滤。我们证明了这种过滤的索引集包含整数的自然数作为一个有序子集。
In this paper we provide a new obstruction to 0-concordance of knotted surfaces in $S^4$ in terms of Alexander ideals. We use this to prove the existence of infinitely many linearly independent 0-concordance classes and to provide the first proof that the submonoid of 2-knots is not a group. The main result is that the Alexander ideal induces a homomorphism from the 0-concordance monoid $\mathscr{C}_0$ of oriented surface knots in $S^4$ to the ideal class monoid of $\mathbb{Z}[t^{\pm1}]$. Consequently, any surface knot with nonprincipal Alexander ideal is not 0-slice and in fact, not invertibl…
A new invariant of equivariant concordance and results on 2-bridge knots
该研究聚焦2-桥结的等变配边类，给出蝶形多项式计算公式、证明无等变切片2-桥结，还提出强可逆结的新等变配边不变量，借此证明2-桥结等变配边阶为无穷。
Structure in the classical knot concordance group
我们提供了新的信息结构的阿贝尔群的拓扑协调类结$S^3$。一个结果是，有一个子群的无限秩完全由结消失卡森戈登不变量，但其非平凡检测$L^{(2)}$签名。
A Second Order Algebraic Knot Concordance Group
我们定义了一个代数群，包括对称的链复形，它捕获的前两个阶段的Cochran-Orr-Teichner可解过滤的结和谐组在一个单一的障碍。为了实现这一点，我们在每个链复形上施加额外的结构，这对基本群进行了额外的控制，特别是对它们可以一致地改变的方式进行了控制。
Given a fixed knot P in a solid torus and any knot K in S^3, one can form the satellite of K with pattern P. This operation induces a self-map of the concordance group of knots in S^3. It has been proved by Dai, Hedden, Mallick, and Stoffregen that in the smooth category there exist P for which this function is rank-expanding; that is, for some K, the set {P(nK)} generates an infinite rank subgroup. Here we demonstrate that similar examples exist in the case of the topological locally flat concordance group. Such examples cannot exist in the algebraic concordance group.
Kawauchi defined a group structure on the set of homology $S^1$$\times$$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injective. We show that this question has a negative answer in the smooth category. Indeed, using knot concordance invariants derived from knot Floer homology we show that the kernel of the zero-surgery homomorphism contains a $\mathbb{Z}^\infty$-subgroup.
Concordance Crosscap Numbers of Knots and the Alexander Polynomial
对于一个纽结K，协调交叉数c（K）是所有与K协调的纽结中的最小交叉数。在G. Zhang等人研究了c（K）< 2的纽结的行列式，我们应用Alexander多项式构造了c（K）< 2的新的代数障碍。除了先前已知的c（K）< 2的低交叉数结外，障碍适用于所有11个或更少交叉的四个素数结。
The Conway knot has infinite concordance order
利用 Rasmussen 不变量、卫星操作和零调扭转，证明 Conway 结在光滑协和群中具有无限阶。
To a special type of grope embedded in 4-space, that we call an admissible grope, we associate a length function for each real number q at least 1. This gives rise to a family of pseudo-metrics d^q, refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken over all possible grope concordances between two knots. We investigate the properties of these metrics. The main theorem is that the topology induced by this metric on the knot concordance set is not discrete for all q>1. The analogous statement for links also holds for q=1. In a…
The $0$-surgeries of two knots $K_1$ and $K_2$ are homology cobordant rel meridians if there exists a $\mathbb{Z}$-homology cobordism $X$ between them such that the two knot meridians are in the same homology class in $H_{1}(X,\mathbb{Z})$. In this paper, we give a pair of rationally slice knots which are not smoothly concordant but whose $0$-surgeries are homology cobordant rel meridians. One knot in the pair is the figure eight knot, which has concordance order two; all previous examples of such pairs of knots are infinite order.
Ribbon concordance and cabling
本文提出关于 ribbon concordance 与缆化纽结的猜想，证明其在特定情形成立，运用纽结 Floer 同调的最小高度不变量，还得到相关亏格界，为纽结理论研究提供新结论。
Splitting the concordance group of algebraically slice knots
作为一个推论的工作Ozsvath和Szabo [数学。GT/0301149]，它表明，经典的协调组的代数切片结有一个无限循环和，特别是不是一个可除群。
A note on the involutive concordance invariants for certain (1,1)-knots
该研究完成了10交叉和11交叉(1,1)-纽结的对合协变不变量的计算工作，为相关纽结理论研究提供了具体的计算结果支撑。
Khovanov concordance minima and the (4,5) torus knot
研究了 (4,5) 螺旋结的 Khovanov homology 在 concordance 类中的性质，证明其为该类中任意结的 Khovanov homology 的 summand。
For all n > 0 there is a homomorphism from the smooth concordance group of knots in dimension 2n + 1 to an algebraically defined group called the rational algebraic concordance group. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n > 1 the homomorphism is injective. This leads to what is called a primary decomposition theorem for knot concordance. In the classical dimension, the kernel of this homomorphism includes the smooth concordance group of topologically slice knots, and Jae Choon Cha has begun studying possible primary decompositions of th…
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.
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.
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