Turaev genus, knot signature, and the knot homology concordance invariants
Oliver T. Dasbach, Adam M. Lowrance
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。
We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.
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We show that the differences between various concordance invariants of knots, including Rasmussen's $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the first example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar-Kofman.
Almost alternating knots with 12 crossings and Turaev genus
在\cite{1}中给出的具有$n\le 12$交叉的非交替结中，Turaev属的191个结未知。对于其中的154个，我们证明它们几乎是交替的，所以它们的Turaev亏格是1。
The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link…
We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple closed curves on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm.
A survey on the Turaev genus of knots
纽结的Turaev亏格是给定纽结距离交替多远的拓扑度量。最近的工作由几个作者集中注意力在这个有趣的不变量。我们讨论了Turaev属是如何与其他结不变量，包括琼斯多项式，结同调论，带状图多项式不变量。
Near extremal Khovanov homology of Turaev genus one links
本文研究 Turaev 亏格一链环在 Khovanov 同调非平凡的前后两个多项式分次中的结构，证明其中一个特定直和项平凡，并由此计算 Rasmussen s 不变量并给出光滑四亏格上界。
A characterization of adequate Turaev genus one links
本文给出恰当链环 Turaev 亏格为一的等价刻画：其 Jones 多项式跨度恰等于交叉数减一。
Invariants for Turaev genus one links
本文研究Turaev亏格一链环的不变量，证明其符号差可由全A Kauffman状态分支数、正交叉数和行列式确定，并证明其Jones多项式首项或末项系数绝对值必为1。
We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that similar classification theorems exist for all genera.
The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or fewer strand up to an error of at most two. Additional bounds are given on the Turaev genus and dealternating number of torus links with five or fewer strands and on some infinite families of torus links on six strands.
A counterexample to Question 1 of "A survey on the Turaev genus of knots"
在“关于Turaev属的结的调查”中，Champanerkar和Kofman提出了几个悬而未决的问题。第一个问题的多项式的系数计数的拟树的所有A带状图的数量从图的最小Turaev亏格是一个不变量的结。我们回答否定的显示从两个图$8_{21}$在KnotAtlas和KnotScape的反例。
On the arc index and Turaev genus of a link
本文计算了充分链结的弧指数，并界定了正3-辫闭合的弧指数。同时提出了一条关于链结交叉数、弧指数和图尔亚夫亏格的不等式猜想，并验证了多个无限链结家族。
The Stable Concordance Genus
纽结的协调亏格是其协调类中任何纽结的最小亏格。虽然很难计算，但它是一个有用的不变量，突出了三亏格和四亏格之间的区别。本文定义并讨论了纽结的稳定协调亏格，它描述了在连通和下协调亏格的行为。
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 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。
On knot Floer width and Turaev genus
对于每一个纽结$K\subset S^3$，我们可以把它的纽结弗洛尔同调调$\hat{HFK}(K)$，一个可生成的二阶阿贝尔群联系起来。一般来说，这些同调群的非零秩位于关于二分次的有限个斜率为1的直线上。同调的宽度实质上是两条这样的线之间的最大水平距离。此外，对于$K$的每个图$D$，都有一个相关联的Turaev曲面，并且Turaev亏格是$K$的所有Turaev曲面的最小亏格。我们证明了纽结弗洛尔同调的宽度受Turaev亏格加1的限制.给出了生成纽结弗洛尔同调的复形的Turaev曲面的亏格和宽度的Skein关系。
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。
This paper presents a way to estimate the Heegaard genus of a $3$-manifold using the Turaev-Viro state sum TQFT. The Turaev-Viro state sum TQFT is derived from the modular category associated to the quantum group $U_q(\mathfrak{sl}_2)$, which is unitary for some $q$ by Wenzl. Hence by Turaev and Virelizier the corresponding TQFT is unitary. We modify a proof by Garoufalidis to give a lower bound of the Heegaard genus using a unitary TQFT, and then use the software Regina to provide some known counterexamples to the rank versus genus conjecture.
We give a tensorial description of the Turaev cobracket on any genus 0 compact surface through the standard group-like expansion, where the Bernoulli numbers appear.
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.
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.
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。
Concordance of Bing doubles and boundary genus
Cha和Kim证明了，如果一个纽结K不是代数切片，那么K的任何迭代Bing double都不与解链相容。证明了如果K具有非平凡签名$σ$，则K的n-迭代Bing双不与亏格小于$2^{n-1}σ$的边界曲面的任何边界链一致.用$2τ$替换$σ$，得到相同的结果，$σ$是Ozsvath-Szabo结一致性不变量的两倍。
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.
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.
The concordance genus of knots
在纽结协调中自然产生三个属g（K）、g_4（K）和g_c（K）：它们是经典属、四球属和协调属，协调属被定义为所有与K协调的纽结中的最小属。显然0 <= g_4（K）<= g_c（K）<= g（K）。Casson和Nakanishi举例说明g_4（K）不必等于g_c（K）。我们首先回顾和扩展他们的结果。对于代表A中元素的节点，代数切片节点的和谐群，这些属之间的关系不太清楚。Casson和Gordon关于A是非平凡的结果意味着对于A中的纽结g_4（K）可以是非零的。Gilmer证明了对于A中的纽结，g_4（K）可以是任意大的.我们将证明在g_4（K）= 1且g_c（K）任意大的情况下，A中存在纽结K.最后，我们列出了所有有10个交叉点的素数结的g_c，除了两个例外，所有有少于10个交叉点的素数结。这就需要对以前未注意到的一致性进行描述。
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。
Milnor's $\barμ$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our ext…
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.
Given an element in the first homology of a rational homology 3-sphere $Y$, one can consider the minimal rational genus of all knots in this homology class. This defines a function $Θ$ on $H_1(Y;\mathbb Z)$, which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function using the correction terms in Heegaard Floer homology. As a corollary, we show that Floer simple knots in L-spaces are genus minimizers in their homology classes, hence answer questions of Turaev and Rasmussen about genus minimizers in lens spaces.
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.
We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are also presented.
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 define a family of link concordance invariants $\left\{ s_n \right\}_{n=2,3, \cdots}$. These link concordance invariants give lower bounds on the slice genus of a link $L$. We compute the slice genus of positive links. Moreover, these invariants give lower bounds on the link splitting number of a link. Especially, this new lower bound determines the splitting number of positive torus links. This is a generalization of Lobb's knot concordance invariants $\left\{ s_n \right\}$, obtained from Gornik's spectral sequence.
Quantum invariants can provide sharp Heegaard genus bounds
利用Boileau和Zieschang的Seifert纤维三流形的例子，我们证明了Reshetikhin-Turaev量子不变量可以用来提供一个严格大于基本群秩的Heegaard亏格的精确下界。
In this paper, we describe a surprising link between the theory of the Goldman-Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara-Vergne (KV) problem in Lie theory. Let $Σ$ be an oriented 2-dimensional manifold with non-empty boundary and $\mathbb{K}$ a field of characteristic zero. The Goldman-Turaev Lie bialgebra is defined by the Goldman bracket $\{ -,- \}$ and Turaev cobracket $δ$ on the $\mathbb{K}$-span of homotopy classes of free loops on $Σ$. Applying an expansion $θ: \mathbb{K}π\to \mathbb{K}\langle x_1, \dots, x_n \rangle$ yields an algebraic description of the operatio…
We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\rm KV}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra $\mathfrak{g}^{(g, n+1)}$. In more det…
For an oriented 2-dimensional manifold $Σ$ of genus $g$ with $n$ boundary components the space $\mathbb{C}π_1(Σ)/[\mathbb{C}π_1(Σ), \mathbb{C}π_1(Σ)]$ carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtration) is described by cyclic words in $H_1(Σ)$ and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [G. Massuyeau, Formal descriptions of Turaev's loop operations] using Kontsevich integral…
We construct the concordance invariant coming from the $E(-1)$ spectral sequence on Khovanov homology in the same way Rasmussen's $s$ invariant comes from the Lee spectral sequence, and show that it gives a bound on the nonorientable slice genus.
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.
The $ν^+$-equivalence classes of genus one knots
本文研究纽结协调群上的 $ν^+$-等价，证明任意亏格为1的纽结必 $ν^+$-等价于三叶结、其镜像或平凡纽结。
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。
By considering a version of Khovanov homology incorporating both the Lee and $E(-1)$ differentials, we construct a $1$-parameter family of concordance homomorphisms similar to the Upsilon invariant from knot Floer homology. This invariant gives lower bounds on the slice genus and can be used to prove that certain infinite families of pretzel knots are linearly independent in the smooth concordance group.
Surfaces in 4-manifolds: Concordance, Isotopy, and Surgery
本文证明了单连通4-流形中同一亏格和同调类的两个曲面是协调的。我们将证明，当它们的互补物具有环状基本群时，它们通常是拓扑同位素的。最后，我们将证明，如果它们是0-一致的，那么对其中一个进行手术就等同于对另一个进行手术。
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.
Four-ball genus bounds and a refinement of the Ozsvath-Szabo tau-invariant
基于Rasmussen的工作，我们构建了一个和谐的不变量与结Floer复杂，并展示的例子中，这个不变量给任意更好的界限上的4球属比Ozsvath-Szabo tau不变量。
We introduce a generalization of the Ozsváth-Szabó $τ$-invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus links and we also give an application to Legendrian link invariants in the standard contact 3-sphere.
We prove that for any winding number $m>0$ pattern $P$ and winding number $-m$ pattern $Q$, there exist knots $K$ such that the minimal genus of a cobordism between $P(K)$ and $Q(K)$ is arbitrarily large. This answers a question posed by Cochran-Harvey [CH17] and generalizes a result of Kim-Livingston [KL05].
We define two concordance invariants of knots using framed instanton homology. These invariants $ν^\sharp$ and $τ^\sharp$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $τ$ invariant in Heegaard Floer homology. We use $ν^\sharp$ and $τ^\sharp$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyper…
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.
We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds. Afterwards, we focus the attention on the case of lens spaces, and use a modified version of the Ozsvath-Szabo-Rasmussen's tau-invariant to obstruct almost-concordances and prove that each L(p,1) admits infinitely many nullhomologous non almost-concordant knots. Finally we …
Genus, Fiberedness, $τ$ and $ε$ of Satellite Knots with $n$-Twisted Generalized Mazur patterns
本研究针对一类推广Mazur模式的(1,1)-模式纽结，计算其n-扭卫星结的协变不变量，证明相关模式的非满射性，确定纤维化条件与亏格，并验证对应卫星结非Floer薄，丰富了卫星纽结的拓扑性质研究。
Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants $TV_r$ of hyperbolic 3-manifolds. We show that if the $r$-growth of $|TV_r(M)|$ for a hyperbolic 3-manifold $M$ that fibers over the circle is exponential, then the monodromy of the fibration of $M$ satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constru…
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.
String topology coproduct and Turaev cobracket on surfaces
本文给出高亏格曲面弦拓扑余积的完整算法计算，并证明弦余括号等于 Turaev 余括号的负值。
In an earlier work, we introduced a family of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK-minus. The resulting groups were then used to define concordance homomorphisms indexed by t in [0,2]. In the present work we elaborate on the special case t=1, and call the corresponding modified knot Floer homology the unoriented knot Floer homology. Using elementary methods (based on grid diagrams and normal forms for surface cobordisms), we show that the resulting concordance homomorphism gives a lower bound for the smooth 4-dimensional crosscap nu…
We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. We also provide applications to the geometry of the infinite braid group. In particular, we show that its commutator subgroup admits a stably unbounded conjugation invariant norm. This answers an…
Let $Σ_0,Σ_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset Σ_0 \times [0,1]$ and $K_1 \subset Σ_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a smoothly and properly embedded annulus $A$ in $W \times [0,1]$ such that $\partial W=Σ_1 \sqcup -Σ_0$ and $\partial A=K_1 \sqcup -K_0$. This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $K \subset Σ\times [0,1]$ admits a no…
Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds
本文利用等变Seiberg-Witten-Floer上同调构造了一族新的纽结同痕不变量 $θ^{(q)}(K)$，并证明它们能给出定4-流形中同调平凡光滑曲面的亏格下界，推广了 $ν^+(K)$ 的相关结果。
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.
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