A. Del Pozo Manglano, P. M. G. Manchón
We give a general procedure that provides, given any particular pretzel link, a braid whose closure is the pretzel link. Moreover, we manage to give a specific braid word in terms of the entries of the pretzel link.
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Quasipositivity and braid index of pretzel knots
本文利用Morton-Franks-Williams不等式和Khovanov-Rozansky concordance homomorphisms，计算了一条链索上具有偶数个交叉点的三股pretzel knots的braid index，并确定了其quasipositivity。
The Braid Indices of Pretzel Links: A Comprehensive Study, Part II
本文作为pretzel链辫子指标综合研究的第二部分，针对类型3 pretzel链给出了辫子指标的确定方法，除少数例外可精确确定，例外情形也能限定在两个连续整数内，并猜想实际指标为其中较大的整数。
The Braid Indices of Pretzel Links: A Comprehensive Study, Part I
本文通过基于标准链环图 Seifert 圈分解的系统分类方法，研究并完全确定了所有 Type 1 和 Type 2 非交错 pretzel 链环的精确辫指数。
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。
The rational Khovanov homology of 3-strand pretzel links
3股椒盐卷饼结和链接是纽结理论中一个经过充分研究的例子来源。然而，虽然已经计算了3股椒盐卷饼结的一些子家族的Khovanov同源性，但没有给出所有这些的通用公式。我们给出了一个一般公式的未约Khovanov同源的所有3链椒盐卷饼链接，在有理数。
The Conway polynomials and Self Delta-equivalence of pretzel links
本文研究带扣链的自delta等价性，针对2组件链给出康威多项式完全不变量，计算其值；对3个或更多组件链，给出自delta等价的充要条件。
A Generalization of Pretzel Links via Spatial Graphs
本文通过空间图推广pretzel链环，构建无限多个不同ribbon knots，并利用jones多项式区分具有相同alexander多项式的链环。
The Enumeration of Alternating Pretzel Links
本文研究交替扭结链环的计数问题，给出按交叉数 $c$ 计算其总数 $\mathcal{P}(c)$ 的闭式公式，并通过数值计算揭示其指数增长规律。
The Thurston polytope for four-stranded pretzel links
本文利用Heegaard Floer环的同调，确定了P（-2r_1-1，2q_1，-2q_2，2r_2 +1）型Pretzel环的对偶Thurston多胞形，其中ri，qi为正整数.我们应用这一结果来确定Thurston规范的跨越表面的各个环节的组成部分，我们明确构建规范实现这样的表面的例子。
There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S, there is only a finite number of pretzel links whose Jones polynomials have span S. More concretely, we provide an algorithm useful for deciding whether or not a given knot is pretzel. As an application we identify all the pretzel knots up to nine crossings, proving in particular that $8_{12}$ is the first non-pretzel knot.
On $χ-$slice pretzel links
本文研究扭结链环的 $χ-$切片性，完全分类正、负情形并部分分类3股与4股情形，进而给出界定有理同调四维球的无穷多族Seifert纤维空间。
Pretzel links, mutation, and the slice-ribbon conjecture
本文利用3重分支覆盖和Donaldson对角化定理证明2分量pretzel link P(p,q,-p,-q)不是slice的，并由此证明了4股2分量pretzel links的slice-ribbon猜想。
The colored $\mathfrak{sl}_{3}$ Jones polynomial $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ are given by a link and an $(n_{1}, n_{2})$-irreducible representation of $\mathfrak{sl}_{3}$. In general, it is hard to calculate $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ for an oriented link $L$. However, we calculate the one-row $\mathfrak{sl}_{3}$ colored Jones polynomials $J_{(n, 0)}^{\mathfrak{sl}_{3}}(P(α,β,γ);q)$ for three-parameter families of oriented pretzel links $P(α,β,γ)$ by using Kuperberg's linear skein theory by setting $n_{2}=0$. Furthermore, we show the existence of the tails …
On geometric realizations of the extreme Khovanov homology of pretzel links
本文通过构造扭结链环实极端 Khovanov 同调的显式几何实现，为 Przytycki–Silvero 关于任意链环图极端 Khovanov 同调无挠的猜想提供新的支持证据。
This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite families of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden symmetries grows linearly with volume. This second family realizes…
Kauffman bracket skein module of the $(3,3,3,3)$-pretzel link exterior
本文证明$(3,3,3,3)$-pretzel链环外部空间的Kauffman括号skein模非有限生成，否证了Detcherry 2021年的有限性猜想。
Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
本文将双辫结的微分展开因式分解研究推广至由三叶结保持缺陷形变得到的零缺陷反平行三 pretzel 结，发现其 DE 系数因式分解虽显著受损，但在对称表示中仍具优雅描述。
The universal character ring of the (-2,2m+1,2n)-pretzel link
我们明确地计算了（-2，2m +1，2n）-pretzel环的泛特征标环，并证明了它对所有整数m和n都是约化的.
Alexander-Conway and Bracket Polynomials of Pretzel Links $\boldsymbol{P(1,1,n)}$
本文研究三股 pretzel 链环 P(1,1,n) 的 Alexander-Conway 与 Kauffman bracket 多项式，重新考察这两个经典多项式不变量，以深化对其拓扑和代数性质的理解。
We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Ayumu Inoue's article a…
Ribbonlength bounds for pretzel links and knots with $\leq 9$ crossings
本文研究了不超过9个交叉的结与结带链的结带长度界限，证明了结带链的最小折叠结带长度存在统一上界，并提供了相关数据表。
Quandle coloring quivers of pretzel links
本文系统研究pretzel链环的二面体quandle着色及着色拟阵，确定3-pretzel情形，并在n为素数时给出4-pretzel及一般m-pretzel链环的着色数和拟阵结构，附完整证明。
We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if $δ_n = σ_1 σ_2 \ldots σ_{n-1}$ is the dual Garside element and $b = δ_n^k P \in B_n$ is a strongly quasipositive braid whose braid closure $\widehat b$ is definite, then $k \geq 2$ implies that $\widehat b$ is one of the torus links $T(2, q), T(3,4), T(3,5)$ or pretzel links $P(-2, 2, m), P(-2,3,4)$. Applying Theorem 1.1 …
Geometric triangulations of a family of hyperbolic 3-braids
本文为$n\ge7$的$(-2,3,n)$-椒盐结与链环补构造拓扑三角剖分，利用Casson和Rivin定理直接证明其几何性，给出了该类双曲3-辫的直接证明。
Twisting quasi-alternating links
对于Khovanov同调和纽结弗洛尔同调，准交替链都是同调稀疏的。我们表明，每一个准交替链接产生一个无限的家庭的准交替的链接，通过更换交叉与交替的理性纠结。因此，我们发现，许多椒盐卷饼链接是准交替的，我们确定的厚度Khovanov同源的“大多数”椒盐卷饼链接任意多股。
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。
Colored HOMFLY polynomials for the pretzel knots and links
借助于演化方法，我们计算了一个庞大的（广义）椒盐卷饼链族的所有对称表示[r]中的所有HOMFLY多项式，该链族由g+1个平行或反平行的两股辫构成，并且依赖于g+1个整数。我们证明了它们具有一个明显的新结构：它们被分解为g+1个初等多项式的乘积之和，这些多项式是借助于重新标度的SU_q（N）Racah矩阵由演化特征值通过旋转得到的，我们给出了一个明确的表达式。广义椒盐卷饼家族包含许多突变体，不可区分的对称HOMFLY多项式，因此，我们的结果扩展到非对称表示R是一个具有挑战性的开放问题。最后，对于具有任意g和R的整个族，可以证明所建议的公式的非平凡推广。
We give an alternate expansion of the colored Jones polynomial of pretzel links which recovers the degree formula in arXiv:1807.00957. As an application, we determine the degrees of the colored Jones polynomials of a new family of 3-tangle pretzel knots.
Character varieties of some families of links
本文考虑了几类环，包括（-2，2m +1，2n）-pretzel环和扭曲Whitehead环。我们计算了这些族的特征标簇，并确定了这些特征标簇的不可约分支的个数。
Satellite fully positive braid links are braided satellite of fully positive braid links
本文研究全正辫子链环的卫星结构，证明其为卫星链环当且仅当模式为含足够多完全扭转的正辫子，并据此刻画平凡纽结。
We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.
Extended welded links are a generalization of Fenn, Rimányi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects.
The volume of positive braid links
基于Futer，Kalfagianni和Purcell最近的工作，我们证明了足够复杂的正辫链的体积与签名缺陷$Δσ=2g-σ$成正比.
This is an expository article on diagrammatic representations of knots and links in various settings via braids.
For a positive braid link, a link represented as a closed positive braids, we determine the first few coefficients of its HOMFLY polynomial in terms of geometric invariants such as, the maximum euler characteristics, the number of split factors, and the number of prime factors. Our results give improvements of known results for Conway and Jones polynomial of positive braid links. In Appendix, we present a simpler proof of theorem of Cromwell, a positive braid diagram represent composite link if and only if the the diagram is composite.
Strongly quasipositive links with braid index 3 have positive Conway polynomial
强拟正链环是那些可以被看作是正辫的带生成元闭包的链环。本文给出了辫指数为3的链环为强拟正的一个必要条件，证明了在这种情况下，链环具有正的Conway多项式（即其所有系数都非负）.我们还表明，这一结果不能扩展到更高数量的股，因为我们提供了一个强准正的辫子，其封闭具有非正的康威多项式。
Exchange moves and braid representations of links
我们证明，在相当一般的条件下，迭代交换移动给无穷多个非共轭辫子。因此，每一个纽结都有无穷多个n-辫表示的共轭类当且仅当它有一个允许交换移动的共轭类。
Braiding surface links which are coverings over the standard torus
我们考虑四维空间中的一个曲面链，它可以由标准环面上的一个简单分支覆盖来表示，我们称之为环面覆盖链。环面覆盖链接包括旋转$T^2$ -结和旋转旋转$T^2$ -结。本文在标准2-球面上编织了一个环面覆盖环。这给出了环面覆盖链环的辫状指数的上估计。特别是，我们表明，转向纺$T^2$-结的环面$(2,\,p)$ -结的编织指数为4。
It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, whi…
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.
Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types
本文通过最小辫子视角研究Lorenz链环与T-链环的等价性，建立了V-链环与T-链环的显式对应，并利用该对应研究T-链环表示的唯一性及几何类型判别。
It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals the braid index $\textbf{b}(D)$ of $D$ if $D$ contains no {\em lone crossings} (a crossing in $D$ is called a {\em lone crossing} if it is the only crossing between two Seifert circles in $D$). If $D$ contains lone crossings, then $\textbf{b}(D)$ is strictly less than $s(D)$. However in general it is not known how $s(D)$ is related to $\textbf{b}(D)$. In this …
We introduce a notion of "quasi-right-veering" for closed braids, which plays an analogous role to "right-veering" for open books. We show that a transverse link $K$ in a contact 3-manifold $(M,ξ)$ is non-loose if and only if every braid representative of $K$ with respect to every open book decomposition that supports $(M,ξ)$ is quasi-right-veering. We also show that several definitions of "right-veering" closed braids are equivalent.
Showing distinctness of surface links by taking 2-dimensional braids
对于一个定向的表面链接$S$，我们可以在$S$上取一个称为2维辫的卫星结构，它是一个覆盖$S$的表面链接。我们证明，二维辫子表面链接是有用的，显示表面链接的独特性。我们研究了具有秩为2的自由阿尔贝群的曲面链环的非平凡例子，得出了它们的链环类型是无穷多的结论。
Coloring Link Diagrams And Conway-type Polynomial Of Braids
本文定义并给出了Artin辫群$B_m$中共轭类的三元Laurent多项式不变量的一个简单组合公式。我们证明了这个Laurent多项式满足Conway绞关系，并且它的系数是辫子的Vassiliev不变量。
We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combined with Ozsváth, Stipsicz, and Szabó's Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six.
Singular twisted links and singular twisted virtual braids
本文引入奇异扭曲纽结理论，证明了奇异扭曲链的Alexander定理与Markov定理，并定义了奇异扭曲虚拟辫的幺半群结构及其约化表示。
Twisted virtual braids and twisted links
本文研究扭曲虚拟辫子，证明扭曲链环的 Alexander 型与 Markov 型定理，并给出扭曲虚拟辫子群的群表示及约化表示。
Braid closure union braid axis is ribbon concordance minimal
研究\(S^3\)中带状协调极小纤维化纽结相关问题，核心方法是运用链环弗洛尔同调与经典技术，主要贡献为证明添加辫子闭包可生成带状协调极小链环及相关推论。
Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class
某些环补的基本群的SU（2）-表示簇的上同调与环的Khovanov同调有相似之处。我们从这里开始一个程序来解释这些相似性的可能来源。我们引入了一个辛流形${\mathscr M}$，它的作用是辫群$B_{2n}$保持辛结构。该动作允许将${\mathscr M}$的拉格朗日子流形与每个辫子相关联。一个链环的表示簇可以被描述为这样的拉格朗日子流形的交集，给出链环的辫子表示。我们预计这将在一定程度上解释上述相似之处。
Floer homology, clasp-braids and detection results
本文拓展Martin的辫轴检测结果，基于链环Floer同调给出近辫状链环的拓扑刻画，受近纤维化纽结研究启发，得到Khovanov同调等的多类检测应用。
We examine the conjecture, due to Champanerkar, Kofman, and Purcell that $\text{vol}(K) < 2 π\log \det (K)$ for alternating hyperbolic links, where $\text{vol}(K) = \text{vol}(S^3\backslash K)$ is the hyperbolic volume and $\det(K)$ is the determinant of $K$. We prove that the conjecture holds for 2-bridge links, alternating 3-braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of $K$ exceeds some function of the twist number of $K$.
Seifert circles, crossing number and the braid index of generalized knots and links
本文引入广义交叉和广义Reidemeister型移动，研究与交叉类型无关的链环性质，证明了广义纽结和链环总交叉数与辫指数间的不等式，并适用于虚拟奇链环。
Braiding link cobordisms and non-ribbon surfaces
我们在$S^3 \times [0,1]$中定义了编织链配边的概念，它推广了$\mathbb{R}^4$中的Viro闭曲面编织。我们证明了任意适当嵌入的定向曲面$W \subset S^3 \times [0,1]$在这个特殊位置上与曲面是同向的，并且当$\partial W$已经由闭辫构成时，同向可以取为相对边界.这些曲面与$D^2 \times D^2$中的另一个曲面编织概念密切相关，称为带帽编织曲面，它是Rudolph编织曲面的推广。我们提到了几个应用程序的编织表面与帽，包括使用它们来应用代数技术从辫群研究表面在4-空间，以及构造奇异纤维化光滑4-流形从一个给定的手柄分解。
The Khovanov width of twisted links and closed 3-braids
Khovanov同调是一个二阶Z-模，它是Jones多项式的范畴。Khovanov同调的支集依赖于有限个关于双分次的斜率为2的直线。科瓦诺夫宽度本质上是两条这样的线之间的最大水平距离。我们表明，它是可能的，以产生无限家庭的链接具有相同的Khovanov宽度从链接图满足一定的条件。因此，我们计算所有封闭的3-辫子的Khovanov宽度。
Braid equivalence in 3-manifolds with rational surgery description
在本文中，我们描述了编织等价的结和链接的3流形$M$通过合理的手术沿框架链接$S^3$。我们首先证明了一个尖锐的版本的Reidemeister定理的链接在$M$。然后，我们通过混合辫子在$S^3$使用$L$ -移动和辫子带移动的辫子等价的几何公式。最后，我们给出了混合辫群$B_{m,n}$的代数公式，使用电缆和技术的混合辫的分梳。当$M$为透镜空间、Seifert流形或由三叶形得到的同调球面时，我们也给出了辫子等价的具体公式.通过混合辫对$3$ -流形的纽结和链环进行代数分类，是研究$3$ -流形的绞模的一个有用工具.
We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.
On chain link surgeries bounding rational homology balls and $χ$-slice 3-braid closures
本文利用格论立方性障碍，确定圆形链环整数手术界定有理同调球的条件，并证明切片-丝带猜想的推广对几乎所有拟交错3辫子链环成立。
Properties of closed 3-braids
我们发现，3-辫子链接与给定的（非零）亚历山大或琼斯多项式是多个，可以有效地确定。我们在封闭的3-辫子强拟正和纤维的，并表明3-辫子链接有一个唯一的不可压缩的塞弗特表面之间进行分类。我们还对Morton-Williams-Franks界为3的正辫子词进行了分类，证明了辫子指数为3的闭正辫子是闭正3-辫子。对于更多的字符串上的封闭辫子，我们研究交替链接发生。特别地，我们对辫子指数为4的那些进行了分类，并证明了它们的Morton-Williams-Franks不等式是精确的。最后，我们使用的Burau表示，以获得新的辫子指数的标准，包括一个有效的4-辫子测试。
The skein polynomial of closed 3-braids
利用3-股辫群的带表示，证明了3-辫链的亏格可以从它们的绞链多项式中读出。给出了一些应用，特别是Morton不等式的一个简单证明和判定某些多项式（如9_{49}的多项式）不被3-辫环所接纳的一个条件.最后，对辫状指数为3的交错链环进行了分类.
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