Sebastian Baader
We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.
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Positive braids of maximal signature
我们通过有限数量的禁止子式来构造具有正Seifert形式的正辫链。由此，我们推导出一个一对一的对应关系之间的总理积极的辫子链接与积极的塞弗特形式和简单的花边Dynkin图，以及一个简单的分类交替积极的辫子结。
We provide the optimal linear bound for the signature of positive four-braids in terms of the three-genus of their closures. As a consequence, we improve previously known linear bounds for the signature in terms of the first Betti number for all positive braid links. We obtain our results by combining bounds for positive three-braids with Gordon and Litherland's approach to signature via unoriented surfaces and their Goeritz forms. Examples of families of positive four-braids for which the bounds are sharp are provided.
We provide linear lower bounds for the signature of positive braids in terms of the three genus of their braid closure. This yields linear bounds for the topological slice genus of knots that arise as closures of positive braids.
On the signature of a positive braid
本文证明正辫子链接的签名下界为其第一贝蒂数的四分之一，改进了费勒先前提出的八分之一下界。
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。
Closures of positive braids and the Morton-Franks-Williams inequality
本文研究了单辫子（也称为正置换辫子）闭包的Morton-Franks-Williams不等式。这允许证明，在一个简单的方式，这套简单的辫子是一个规范正交基的内积的赫克代数的辫群的卡尔曼，谁首先得到了这一结果，通过使用一个有趣的联系与接触拓扑。我们还介绍了一种新的技术来研究的Homflypt多项式的封闭的积极辫子，即决议树的叶子是简单的辫子。在这些简单的决议树，我们刻画了封闭的积极辫子，莫顿-弗兰克斯-威廉姆斯不等式是严格的。特别是，我们明确地确定了积极的辫子字的三股封闭编织指数为3。
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.
On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks
我们给出了一些不寻常的性质的交叉数的正图或股数的正辫表示的结的例子。特别是，我们表明，积极的编织结可能没有积极的最小（股数）编织表示，给对方的结果，弗兰克-威廉姆斯和Murasugi。其他的例子回答了克伦威尔关于齐次和（部分）亚当斯关于几乎交替结的问题。我们给出了一个反例，并纠正版本的琼斯定理的亚历山大多项式的4-辫子结。我们还给出了一个结的例子，所有以前应用的编织指数标准未能估计急剧（从下面）的编织指数。之间的关系（推广）这样的例子和琼斯的猜想，最小的辫子表示有独特的writhe的讨论。最后，我们给出了一个反例莫顿猜想有关的亏格和次数的绞多项式。
Non-braid positive hyperbolic $L$-space knots
本文研究非辫正超流体L-空间结，构造了无限多个此类结，证明其非辫正性，与Baker和Kegel的例子不同。
We show that the difference between the Seifert genus and the topological 4-genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot. We deduce that among prime positive braid knots, the property of having such a genus difference less than any fixed constant is characterised by finitely many forbidden surface minors.
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.
The bridge index and the braid index for twist positive knots
本文受Krishna和Morton猜想启发，利用结Floer挠阶证明了所有扭转正结的桥指数与辫指数相等。
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 are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case it has positive Conway polynomial (that is, all its coefficients are non-negative). We also show that this result cannot be extended to a higher number of strands, as we provide a strongly quasipositive braid whose closure has non-positive Conway polynomial.
Braid positive surgery diagrams
研究证明闭合定向连通3流形可通过辫正链环的Dehn手术得到，核心方法是辫正链环的构造与性质分析。
A knot $K_1$ is said to be Gordian adjacent to a knot $K_2$ if $K_1$ is an intermediate knot on an unknotting sequence of $K_2$. We extend previous results on Gordian adjacency by showing sufficient conditions for Gordian adjacency between classes of positive braid knots through manipulations of braid words. In addition, we explore unknotting sequences of positive braid knots and give a proof that there are only finitely many positive braid knots for a given unknotting number.
Conjugacy for positive permutation braids
n弦上的正置换辫子是指每对弦最多交叉一次的正n-辫子，它是辫群共轭性研究中的基本但非平凡的基础.我们考虑了这些基本辫线之间的共轭性，并证明了这些基本辫线与平凡辫线或三叶辫线之间的共轭性。所有这些具有最大可能交叉数的n-辫子也被证明是共轭的。我们注意到，当n<6时，这些辫子的共轭性仅取决于交叉数。相比之下，我们展示了两个这样的辫子6串9交叉不共轭，但其每个封闭的同位素（2，5）环面结。
Satellite knots that cannot be represented by positive braids with full twists
该研究证明了部分伴随与模式均为正辫子闭包的卫星纽结无法用带全扭转的正辫子表示，进而找到无穷多类伴随和模式均为Lorenz纽结但本身非Lorenz纽结的卫星纽结，解答了相关经典问题。
The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N\times N$ symmetric integer matrices with zero diagonal--in fact, it represents the abelianization of this subgroup. As a function on the whole $N$-braid group, it is a derivation defined by the action of the symmetric group on square matrices. The set of all crossing matrices ca…
The depth of a link measures the minimum height of a resolving tree for the link whose leaves are all unlinks. We show that the depth of the closure of a strictly positive braid word is the length of the word minus the number of distinct letters.
We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots.
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.
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。
On unknotting fibered positive knots and braids
本文研究了纤维正结的解结性质，证明了正辫图中存在特定交叉点可解结，并质疑了Stoimenow关于纤维正结的猜想。
Ordering the braid groups
给出了Artin辫群$B_n$是右序群的一个明确的几何证明.该结构是基本的，自然的，并导致一个新的，有效的计算，标准形式的辫子，我们称之为左一致的标准形式。在我们的顺序中为正（分别为负）的辫子的左一致形式在出现的最小辫子生成器中具有一致的正（分别为负）指数。由此可见，我们的排序是相同的Dehornoy，建造非常不同的手段，我们恢复Dehornoy的主要定理，任何辫子可以把这样一种形式使用正或负指数在最小的发电机，但不是两者兼而有之。我们的定义秩序是强有力的连接与莫舍的正常形式，这导致了一个算法，以决定是否一个给定的辫子是积极的，平凡的，或消极的是二次的长度辫子字。
We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element $δ$ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive Hopf bands to a disk. Rudolph constructed fibered strongly quasipositive links as closures of positive words on certain generating sets of $B_n$ and we prove that Rudolph's condition is equivalent to ours. Finally, we show that the braid index is a strict upper bound for the num…
Positive knots have negative signature
J.Birman、Williams和L.Rudolph提出了一个问题，即非平凡的洛伦兹结是否总是具有正的签名。洛伦兹结是正辫的例子（在我们的约定中，它们的交叉点都是负的，所以它们是负链）。鲁道夫（L.Rudolph）证明了正辫子具有正签名（如果它们代表非平凡链接）。K.Murasugi证明了非平凡的、交替的、正的链接有负的签名。在这里，我们证明了一般的老民间传说猜想，非平凡的积极联系有负签名。
We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r < 2g(K)-1$, the 3-manifold obtained via $r$-framed Dehn surgery along $K$ admits a taut foliation. Our main result provides some positive evidence towards this conjecture: we construct taut foliations in such manifolds whenever $r<g(K)+1$. As an application, we produce a novel braid positivity obstruction for cable knots by proving that the $(n,\pm 1)$-cable of a k…
The dealternating number and the alternation number of a closed 3-braid
给出了闭3-辫子的去交错数的一个上界。作为应用，我们确定了一些闭正3-辫的去交错数、交错数和Turaev亏格。我们还证明了存在无穷多个具有任意去交错数（或任意交错数）和任意辫指数的正纽结。
We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot $P(-2,3,7)$, and indeed along every pretzel knot $P(-2,3,q)$, for $q$ a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. …
Taking the Levine-Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017 Cimasoni and Conway generalized this formula to the multivariable signature of the closure of coloured tangles. In the present paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the …
Census L-space knots are braid positive, except for one that is not
该研究证明了除一个例外外，人口普查中的所有L空间结都是正辫结，并探讨了相关多项式性质及半群理论问题。
On the kinkiness of closed braids
本文证明了一个闭辫的正扭结性的下界，并利用它导出了一个链节在其Seifert系统下的正扭结性的估计。作为一个应用程序，我们表明，某些椒盐卷饼结不能解开只使用积极的交叉变化。我们还描述了一个子群的无限秩的光滑纽结和谐群，其中没有任何元素有一个强拟正的代表。
Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considering the multivariable signature of the closure of a colored tangle. The corresponding defect is expressed in terms of the Maslov index and of the Lagrangian functor defined by Turaev and the first-named author.
We use Ozsváth, Stipsicz, and Szabó's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a positive full-twist realize the braid index of their closure. We also establish that quasi-positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure. Finally, we provide inductive formulas for the Upsilon invariant of torus knots and compare it to t…
Every Plat Presentation Admits a Positive Bounded Braid Representative
该研究针对纽结分类中Hilden双陪集类庞大的问题，证明每个Hilden双陪集存在有界正辫子代表元，为桥有限性猜想研究提供了代数方法的建设性工具。
Knot Floer homology of positive braids
研究正辫子闭包链环的纽结弗勒同调次顶层项，证明该族素纽结秩为1且单值化无不动点，与其他有此属性的纽结类比较，给出非正辫子的相关纽结例子并提出后续研究问题。
Positive knots and knots with braid index three have property_p
证明了3-球面上的正纽结和辫指数为3的纽结满足性质P猜想。
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。
Positive presentations of the braid groups and the embedding problem
发现并研究了辫群的一类正有限表示。它表明，没有介绍，但已知的例外，在这类具有的属性，相当于辫子的话也是等价的正关系。
We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot $K$ with property (D) with slope $\frac{p}{q}\ge 2g(K)-1$ is not left orderable. By making full use of the fixed point method, we prove that (1) nontrivial knots which are closures of positive $1$-bridge braids have property (D); (2) L-space satellite knots, with positive $1$-bridge braid patterns, and companion with property (D), have property (D); (3) the fundamental group of the manifold obtained by Dehn filling on $v2503$ is not left orderable. Additionally…
Simple Braids Tend toward Positive Entropy
本文证明当股数趋于无穷时，简单辫子中具有正拓扑熵的比例趋近于100%，此类辫子为伪阿诺索夫或含伪阿诺索夫分量，证明采用约化为非简单3股辫子的方法。
Positive braid closures and taut foliations
该研究探讨了正编织闭包在 $S^3$ 中的紧叶面性质，证明了特定 Dehn 手术条件下可得到非 L-space 并支持 co-oriented 紧叶面。
Positive braids minimize ascending number
本文证明正辫闭包所得结的上升数等于解结数，并通过数据分析探讨正结及低交叉数双曲交替结的上升数在极小交叉数图中实现的性质。
Homogeneous links and closed homogeneous braids
任何正结都是正辫的闭合吗？不，但是如果我们考虑正性，根据伯尔曼、柯和李提出的辫群的生成元，那么答案是肯定的。在本文中，我们证明了同样的情况下，考虑均匀性。在这种方式中，我们证明了两个曲面的铅垂是BKL-齐次曲面当且仅当两个和项都是BKL-齐次曲面，一个平行的结果鲁道夫涉及拟正曲面。BKL-齐次曲面实际上是拟正曲面的自然推广。
Twist positivity, L-space knots, and concordance
本文证明扭正纽结的编织指数出现在 Alexander 多项式第三指数位，并据此证明扭正 L-空间纽结的编织指数等于桥指数，为切片-带状猜想的重释及相关新猜想提供证据。
Satellite Links with Multiple Full Twists and Single-Twist Companions
研究卫星链环的正辫表示中全扭转数与其伴随链环的关系，构造了无限多卫星链环，其正辫表示具有任意多个全扭转，而伴随链环的正辫表示最多只有一个全扭转。
Positive knots, closed braids and the Jones polynomial
利用Polyak-Viro-Fiedler的Vassiliev不变量的Gauss图公式，结合Bennequin不等式，证明了正纽结的Vassiliev不变量与Jones多项式的亏格和次数之间的几个不等式.因此，我们证明了对于Alexander/Conway，Jones，HOMFLY，Brandt-Lickorish-Milkor-Ho和Kauffman的任何多项式，同一多项式只有100多个正纽结，平凡多项式没有正纽结.本文还讨论了Bennequin不等式的一个推广，证明了正纽结的解纽结数不小于它的亏格，恢复了A 'Campo，Kawamura和Tanaka最近的一些解纽结数结果，并给出了正纽结的Jones多项式的应用.
We associate an open book with any connected plane checkerboard graph, thus providing a common extension of the classes of prime positive braid links and positive tree-like Hopf plumbings. As an application, we prove that the link type of a prime positive braid closure is determined by the linking graph associated with that braid.
Simple Braids
本文研究了无平方正辫的一个子集，给出了它的一些代数刻画和一个几何刻画：闭包为非链的正辫集。我们描述这些辫子和共轭类的规范形式。
We study some special systems of generators on finite groups, introduced in previous work by the first author and called "diagonal double Kodaira structures", in order to investigate non-abelian, finite quotients of the pure braid group on two strands $\mathsf{P}_2(Σ_b)$, where $Σ_b$ is a closed Riemann surface of genus $b$. In particular, we prove that, if a finite group $G$ admits a diagonal double Kodaira structure, then $|G|\geq 32$, and equality holds if and only if $G$ is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two $3$-dimen…
On Positive Braids, Monodromy Groups and Framings
本文为每个正辫子构造辫子单值群，推广孤立平面曲线奇点的几何单值群，并在闭包为纽结时将其等同于带框架映射类群，从而得到良定义的纽结不变量，进而证明不可约奇点的几何单值群由亏格和Arf不变量决定。
On T-positive links
本文研究T正链接，证明其为强拟正链接的子集，且包含所有非分裂辫正链接。通过分析T正链接的性质，补充了Rudolph和Banfield对T正链接的先前描述，并探讨了其在缆绕操作和连通和下的行为。
Positive Artin Presentations
本文分类了三维球面中描述正Artin展示的闭合纯n-辫，证明此类辫强可逆，并举例说明存在基本群不容许正Artin展示的三维流形。
Slice-torus link invariants, combinatorial invariants, and positivity conditions
本文证明了链环为拟正、正或正辫子闭包的必要条件，并据此刻画了解链数为1和2的正链环，给出了检验正链环是否为正辫子闭包的组合判据，并编制了交叉数小于8的正素链环表。
Positive Braid Links with Infinitely Many Fillings
该研究证明了非同痕于标准有限型链环的正辫Legendrian链环存在无穷多精确拉格朗日填充，推进了相关拓扑领域研究。
Quasipositive Braids, Presentations, and Garside Structures
本文给出辫子群的一种特定表示，证明其正幺半群不与拟正辫子幺半群同构，且拟正辫子幺半群无真Garside族，解决了Calegari的一个问题。
Positive 3-braids, Khovanov homology and Garside theory
本文利用Garside理论对正3辫共轭类进行分类，并计算其闭包Khovanov同调表的前四列和最低三行，进一步通过共轭最大化下确界以在线性时间内确定更多同调单元。
On asymmetric hyperbolic L-space knots of braid index four
本文推广了已知唯一的辫指数4的非对称L空间纽结t12533，构造出辫指数4的非对称双曲L空间纽结的第一个无限族。
The unknotting number of a positive braid with n strands and k intersections is known to be equal to (k-n+1)/2. We consider Lorenz knots (which are positive braids) and, using a different method, find their unknotting numbers in terms of their positions on the Lorenz attractor.
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