Canonical genus and the Whitehead doubles of pretzel knots
Mark Brittenham, Jacqueline Jensen
我们证明了对于一个交替的椒盐卷饼结K，它的Whitehead doubles的典型亏格W（K）等于K的交叉数c（K），从而验证了Tripp的一个猜想.
We show that for an alternating pretzel knot K the canonical genera of its Whitehead doubles W(K) are equal to the crossing number c(K) of K, verifying a conjecture of Tripp in the case of these knots.
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The JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one
本文研究沿亏格一经典扭结作0-手术所得3-流形的JSJ分解，并借助最小扭曲五链环异常填充的分类结果进行分析。
Some invariants of pretzel links
我们证明了非平凡的经典椒盐卷饼结L（p，q，r）是双曲的，有八个例外是环面结。我们发现康威多项式的n-pretzel链接使用一个新的计算树。作为应用，我们使用这些多项式计算n-pretzel链接的属，并通过证明一个pretzel链接的属和典型属是相同的来找到pretzel链接的篮子数。
A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2,5) and (2,7)-torus knots, the genus 2 and 3 alternating odd pretzel knots do not admit any chirally cosmetic surgeries. Further, we show that for a fixed genus, at most finitely many alternating odd pretzel knots admit chirally cosmetic surgeries.
Crossing number of an alternating knot and canonical genus of its Whitehead double
特里普在2002年提出的一个猜想指出，任何纽结的交叉数与其怀特黑德对偶的典型亏格一致。在此期间，它已经建立，这一猜想是正确的一大类交替结，包括$(2, n)$环面结，$2$ -桥结，代数交替结，交替椒盐卷饼结。本文证明了对于两个$(2, m)$型和$(2, n)$型环面纽结的连通和为交错$3$ -辫子纽结的情形，上述猜想不成立.这导致了一个新的修正猜想，即任何素数纽结的交叉数与其Whitehead double的典型亏格一致。我们还给出了一个新的满足猜想的大类素数交错纽结，包括所有素数交错$3$ -辫子纽结.
The canonical genus for Whitehead doubles of a family of alternating knots
对任意给定的整数$r \geq 1$和一个具有$ε=\pm 1$的拟序辫子$β_r=(σ_r^{-ε} σ_{r-1}^ε...$ $ σ_{1}^{(-1)^{r}ε})^3$，证明了闭包$\hatβ_r$的双链$W_2(\hatβ_r)$的HOMFLYPT多项式$P_{W_2(\hatβ_r)}(v,z)$在$z$中的最大次数等于$6r-1$ .作为应用，我们给出了交替纽结族$\mathcal K^3$，包括$(2,n)$环面纽结、2-桥纽结和交替椒盐卷饼纽结作为它的子族，使得$\mathcal K^3$中任意交替纽结的最小交叉数与它的Whitehead二重化的典型亏格一致.因此，我们给出了一个新的家庭$\mathcal K^3$的交替结特里普猜想成立。
We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy representations determine the genus and fiberedness of the knots.
Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots
本文利用 Mayland 技术证明若干亏格一及高亏格环面链环结的纽结群双可序，首次给出非纤维化或非交错纽结的双可序例子。
Colored knot polynomials for Pretzel knots and links of arbitrary genus
一个非常简单的表达式是任意有色琼斯和HOMFLY多项式的丰富的$(g+1)$参数家庭的椒盐节和链接。Jones多项式和HOMFLY多项式的解通过U_q（SU_N）的Racah矩阵完全而明确地表达出来，并且看起来与复曲面保形块的模变换有关。
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。
Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots
本文基于之前的工作，将构造正允许Lefschetz纤维的方法应用于交替结和扩展交替结的结迹，证明其纤维的亏格等于平面图中的白区域数，并得出正预结结和正扭转结的结迹的纤维亏格结果。
Cosmetic crossings of genus one knots
我们发现，对于属一节的亚历山大多项式和同源的双重覆盖分支结提供障碍化妆品交叉。作为应用，我们证明了所有纽结的负扭正Whitehead双纽结的无效穿越猜想。我们还验证了几个家庭的椒盐卷饼结和所有属一个结多达10个交叉的猜想。
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.
Tunnel number one, genus one fibered knots
我们确定亏格1纤维结的透镜空间，隧道数为1。我们还表明，每一个隧道数1，一旦穿孔环面束的结果，德恩填充组件的Whitehead链接在3球。
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant.
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。
Longitude Floer homology and the Whitehead double
我们定义了S ^3中纽结K的经度弗洛尔同调，并证明了它是K的一个拓扑不变量。得到了这些同调群的一些基本性质。特别是，我们表明，他们区分属K。我们还对（2，2n+1）环面结进行了显式计算。最后得到了K的经度弗洛尔同调与其Whitehead双K_L的Ozsvath-Szabo弗洛尔同调之间的对应关系。
Cosmetic crossings and Seifert matrices
我们研究化妆品交叉结属1，并获得障碍，这种交叉的结不变量决定的塞弗特矩阵。特别是，我们证明，对于属一节的亚历山大多项式和同源的双盖分支结提供障碍化妆品交叉。作为应用，我们证明了非索纽结的扭曲Whitehead对偶的无效穿越猜想。我们还验证了几个家庭的椒盐卷饼结和所有属一个结多达12个交叉的猜想。
Lens space surgeries on knots of small genus
本文证明了亏格至多5的纽结中，仅 $(-2,3,7)$ 扭结有透镜空间手术，并通过关键技术结果结合先前工作验证了 Berge 猜想。
We give a criterion for distinguishing a prime knot $K$ in $S^3$ from every other knot in $S^3$ using the finite quotients of $π_1(S^3\setminus K)$. Using recent work of Baldwin-Sivek, we apply this criterion to the hyperbolic knots $5_2$, $15n_{43522}$, and the three-strand pretzel knots $P(-3,3,2n+1)$ for every integer $n$.
Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior
令$δ_g$是最小扩张的pseudo-Anosovs上的一个封闭的表面$Σ_g$的$g$和$δ_g^+$是最小扩张的pseudo-Anosovs上的$Σ_g$与定向不变的叶理。本文研究了幻流形$N$的每个$r \in \{-3/2, -1/2, 2\}$的$N(r)$的Dehn填充在闭纤维上作为单值出现的伪Anosovs.流形$N(-3/2)$同胚于Whitehead姐妹链外环。我们认为集合$Λ_g(r)$（resp.《易经》云：“凡所有相，皆是虚妄。具有可定向不变叶理的单值体）在$g$属的闭合纤维上，用于$N(r)$的Dehn填充物，其中填充物在$N(r)$的纤维的边界斜坡上。Hironaka分别通过计算$\min Λ_g(-1/2)$和$\min Λ^+_g(-1/2)$得到了$δ_g$和$δ_g^+$的上界。我们证明了$\min Λ_g(-3/2)< \min Λ_g(-1/2)$对于$g \equiv 0,1,5,6,7,9 \pmod{10}$和$\min Λ^+_g(-3/2)< \min Λ^+_g(-1/2)$对于$g \equiv 1,5,7,9 \pmod{10}$。这些不等式改进了$g$的$δ_g$和$δ_g^+$的上界。证明了对于每个$r \in \{-3/2, -1/2, 2\}$和每个$g \ge 3$，对于$N(r)$的Dehn填充，在$g $亏格的闭…
Knot Floer homology, genus bounds, and mutation
在以前的论文中，我们介绍了一个集合的分次阿贝尔群$\HFKa(Y,K)$相关的结在一个三流形。本论文的目的是调查这些群体的几个特定的家庭结，包括木下寺坂结和他们的“康威突变体”。这些结果表明，$\HFKa$包含更多的信息比亚历山大多项式和这些结的签名，他们也说明了事实，$\HFKa$检测突变。我们还计算了某些椒盐卷饼结的$\HFKa$，以及具有小交叉数（$n\leq 9$）的结。我们的计算证明，这里考虑的许多结不承认塞弗特纤维手术。
In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a (2, 2q+1)-torus knot K, we demonstrate that there are infinitely many unknots c such that p-twisting K about c yields a twist family, which consists of hyperbolic knots with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus…
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…
Nonembeddable contractible open manifolds arising from Whitehead doubling
本文研究由非平凡纽结与半扭转迭代怀特海德加倍构造的可缩开流形，证明其不可嵌入特定度量空间，完成分类并给出完全不变量，方法可推广至高维情形。
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