Nakisa Ghanbarian
Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.
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An analog of the {K}auffman bracket polynomial for knots in the non-orientable thickening of a non-orientable surface
本文针对非可定向曲面加厚中的伪经典纽结，通过修改交叉点符号与光滑化定义，构造了考夫曼括号多项式的类似物，并证明其是合痕不变量且独立于经典考夫曼括号多项式。
Stably irreducible non-orientable knotted surfaces
本文提出了四维球面中纽结曲面可约性的基本障碍，并据此构造了稳定不可约的不可定向曲面。
We show that the equivariant and non-equivariant non-orientable 4-genus of p-periodic knots may differ, for any choice of p>1. Similar results have previously been obtained for the smooth 4-genus and non-orientable 3-genus of a periodic knot. These stand in contrast to Edmods' acclaimed result by which the equivariant and non-equivariant Seifert genus of a periodic knot agree.
The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami's and Yasuhara's, and give a new lower bound for the slicing number of knot.
In formulating a non-orientable analogue of the Milnor Conjecture on the $4$-genus of torus knots, Batson developed an elegant construction that produces a smooth non-orientable spanning surface in $B^4$ for a given torus knot in $S^3$. While Lobb showed that Batson's surfaces do not always minimize the non-orientable $4$-genus, we prove that they always do minimize among surfaces that share their normal Euler number. We also completely determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which …
We describe an example of a closed orientable 3-manifold with distinct distance three genus two Heegaard splittings. This demonstrates that the constructions of alternate genus two Heegaard splittings of closed orientable 3-manifolds described by Rubinstein and Scharlemann in their 1998 paper Genus Two Heegaard Splittings of Orientable 3-Manifolds, does not yield all alternate genus two splittings and must be augmented.
An analogue of Turaev comultiplication for knots in non-orientable thickening of a non-orientable surface
本文研究了不可定向曲面加厚流形中的伪经典纽结，引入了 Turaev 余乘法的类似物不变量 $Δ$，并由此推导出包括仿射指标多项式类似物在内的多个同伦、同调和多项式不变量。
Non-orientable 4-genus of torus knots
研究环形结T_{5,q}和T_{6,q}的不可定向4-亏格界限，并推导-1手术后环形结的d不变量通用非递归公式。
We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.
The universal order one invariant of framed knots in most S^1-bundles over orientable surfaces
众所周知，自连接是S^3中框架纽结唯一的Z值瓦西里耶夫不变量。然而，对于大多数3-流形，特别是对于可定向曲面F（而不是S^2）上的S^1-丛的全空间，Z值一阶不变量空间是无穷维的。本文给出了可定向的非紧曲面F（非S^2）上S^1-丛的可定向全空间中框架纽结的一阶不变量I的一个显式公式。我们证明了如果F不是S^2或S^1 × S^1，则I是泛一阶不变量，即它区分每两个可以由值在阿贝尔群中的一阶不变量区分的框架纽结.
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 $γ_4(K)$.
For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define $γ_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that $γ^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $γ^0_X$ is a generalization of the non-orientable 4-ball genus. While it is very likely that for given $X$, $γ^0_X$ has no upper bound, it is difficult to show it. In fact, even in the case of $γ^0_{S^4}$, its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that fo…
J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to the equivalence classes of cords attached to the surface-knot, and correspond to the double cosets of the peripheral subgroup of the knot group. In this paper, we classify cords and cords with local orientations attached to (possibly non-orientable) surface-knots. And we classify 1…
In the symplectization of standard contact $3$-space, $\mathbb R \times \mathbb R^3$, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus $0$. We show that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a non-orientable Lagrangian endocobordism must be a positive multiple of $4$. The more restrictive exact, non-orientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do e…
Non-orientable genus of a knot in punctured $\mathbb{C}P ^2$
本文利用纽结符号差与1-手术所得同调球的Heegaard Floer d不变量，建立穿孔nCP²中不可定向零调曲面第一Betti数的下界，并据此构造任意给定下界的纽结。
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。
We investigate when a Legendrian knot in standard contact $\mathbb{R}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we determine completely when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable, and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, in…
An Algorithmic Search for Knots Bounding Möbius Bands in $B^4$
该研究借助大规模计算搜索和扩展算法，研究纽结的光滑不可定向4-亏格，识别出\(S^3\)中交叉数至多14且在\(B^4\)中界定莫比乌斯带的素纽结，扩展了相关已知例子并提供新数据。
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance genus of 533 of the 552 prime 11--crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11--crossing knots known to be slice.
By considering non-orientable surfaces in the surgered manifolds, we show that the 10/3- and -10/3-Dehn surgeries on the 2-bridge knot $9_{27} = S(49,19)$ are not cosmetic, i.e., they give mutually non-homeomorphic manifolds. The knot is unknown to have no cosmetic surgeries by previously known results; in particular, by using the Casson invariant and the Heegaard Floer homology.
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.
The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots
本文综述并结合多种技术，计算11交叉非交替纽结的不可定向4维亏格，并利用二重分支覆盖分析其能否界定 Möbius 带。
Additivity of free genus of knots
我们证明了在连通和下纽结的自由亏格是可加的。
Genus of Embedded Graphs in Orientable Closed Surfaces
提出一种基于2-球面特殊分支覆盖的算法，用于计算图在可定向闭曲面中的最小与最大亏格，并应用于Snarks图计算其可定向亏格和最大亏格。
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.
We describe the genus two knots which admit a genus one, one bridge position. These are divided into several families, one consists of vertical bandings of two genus one $(1,1)$-knots, other consists of vertical bandings of two cross cap number two 2-bridge knots, and the last one consists of genus two tunnel number one satellite knots.
Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that there are topologically slice knots with 4-ball genus equal to one and arbitrarily large concordance genus.
We characterize composite tunnel number one genus two handlebody-knots.
Knots with unique minimal genus Seifert surface and depth of knots
我们描述了一个程序，创建具有唯一的最小亏格塞弗特曲面的双曲结的无限家庭。这些结的一个大的子集具有进一步的属性，即表面不能是结外部的深度为一个叶状的唯一紧凑的叶。
Knots of genus two
本文对亏格为2和亏格为3的纽结图进行了分类，并给出了正纽结、交错纽结和齐次纽结的应用，包括非手性亏格为2的交错纽结、切片纽结和非手性2-几乎正纽结的分类、3-和4-移动纽结的证明以及弱亏格为2的纽结的最大双曲体积的计算.我们还研究了单位根上的链环多项式的值，推广了Jones的稠密性结果。利用这些值，找到了弱亏格的纽结不等式的例子，并将几个结果推广到任意弱亏格。
Genus two Heegaard splittings of orientable three-manifolds
Bonahon-Otal和Hodgson-Rubinstein证明了同一个三维流形（典型的是一个透镜空间）的任意两个亏格为一的Heegaard分裂是同位素的。另一方面，Boileau、Collins和Zieschang证明了某些塞弗特流形具有不同的亏格2 Heegaard分裂。在较早的文件中，我们提出了一种技术比较Heegaard分裂的同一流形，并使用这种技术，推导出唯一性定理的透镜空间分裂作为一个简单的推论。在这里，我们使用类似的技术来检查，在一般情况下，在两个非同位素属两个Heegard分裂相同的3流形比较的方式，特别关注如何相应的超椭圆对合是相关的。
It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $L^2$-signatures bound this invariant from below.
Trisection genus of knot traces
本文分类了三分亏格至多为2的纽结迹，并构造了无限多具有三分亏格3和4的纽结迹，证明了存在无限族纽结迹具有任意大的三分亏格，同时给出了若干著名纽结迹的三分亏格精确值或紧界。
Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.
Among non-alternating knots with $n\le 12$ crossings given in \cite{1} Turaev genus is not known for 191 knot. For 154 of them we show that they are almost alternating, so their Turaev genus is 1.
We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $δ$-gradings $k$ and $-k$.
We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.
We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the negatively twisted, positive Whitehead doubles of all knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 10 crossings.
Free genus one knots with large volume
一个纽结K的塞弗特曲面F是自由的，如果F的补曲面是一个凸体（即，具有自由基本群）。K的自由亏格是K的所有自由Seifert曲面中的最小亏格。本文证明了存在任意大体积的双曲纽结族，且每个纽结族都有自由亏格1。这意味着，有结的自由属一，但任意大的典型属，并存在结承认不可压缩的自由塞弗特表面不能建立应用塞弗特算法的投影结。
Milnor's invariants for knots and links in closed orientable 3-manifolds
该研究将原本针对三维球面$S^3$中链环的Milnor $\bar{\boldsymbol{\nu}}$-不变量推广到一般闭可定向3-流形的纽结与链环场景，证明其为拓扑协边不变量，统一并拓展了三维下的各类Milnor不变量。
Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4-manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the sim…
A presentation for the mapping class group of the closed non-orientable surface of genus 4
对于任意可定向的紧致曲面F，已知映射类群M（F）的有限表示。如果F是不可定向的，那么只有当F的亏格至多为3且边界分量很少时，才知道这种表示。本文从亏格4的闭非可定向曲面在所谓的曲线序复形上的作用出发，得到了它的映射类群的有限表示。
We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.
In this paper, the support genus of all Legendrian right handed trefoil knots and some other Legendrian knots is computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genus with arbitrarily negative Thurston-Benniquin invariant. This answers a question in Onaran.
Divide knots of maximal genus defect
该研究构造了具有任意光滑四维亏格但拓扑四维亏格为一的分隔结，揭示了强拟正则纤维化结的四维亏格比值可趋近于零的特性。
The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants.
We prove the cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial. We also prove the conjecture for genus one knots with trivial Alexander polynomial, under some additional assumptions.
We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto and Baker.
We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the triple-crossing numbers of all torus knots, and of many more knots and their connected sums.
We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.
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。
The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.
A census of genus two 3-manifolds up to 42 coloured tetrahedra
我们改进和扩展到不可定向的情况下，Karabas，Malicki和Nedela最近的结果关于分类的所有定向总理3-流形的Heegaard属2，三角形最多42个彩色四面体。
Let $N_g$ be the non-orientable surface with genus $g$, $\text{MCG}(N_g)$ be the mapping class group of $N_g$, $\mathcal{T}(N_g)$ be the index 2 subgroup generated by all Dehn twists of $\text{MCG}(N_g)$. We prove that for odd genus, $\mathcal{T}(N_g)$ can be generated by three elements of finite orders.
We give an obstruction for genus one knots $K$, $K'$ to have the Gordian distance one by using the $0$th coefficient of the HOMFLT polynomials. As an application, we give a new constraint for genus one knot to admit a (generalized) cosmetic crossing. Combining known results, we prove the (generalized) cosmetic crossing conjecture for genus one pretzel knots.
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.
We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.
Closed incompressible surfaces of genus two in 3-bridge knot complements
本文刻画了亏格为2的闭不可压缩曲面在3-桥纽结和环的补中的性质。这个特征包括3桥结和链接的基本2弦缠结分解。
A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Naturally the exterior of a 1-genus 1-bridge knot K allows a genus 2 Heegaard splitting, i.e., K has an unknotting tunnel. But, in general, there are unknotting tunnels which are not derived form this procedure. Some of them may be levelled with the torus \partial V_1=\partial V_2, who…
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