Maciej Borodzik, Jakub Zarzycki
We show that under a precise condition on the single variable Alexander polynomial, the limit at $1$ of the Tristram--Levine signature of a link is determined by the linking matrix.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that the maximum of the Levine-Tristram signature function of a torus knot satisfies a reduction formula analogous to a result by Gordon-Litherland-Murasugi for the classical signature.
We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants.
The Levine-Tristram signature associates to each oriented link $L$ in $S^3$ a function $σ_L \colon S^1 \to \mathbb{Z}.$ This invariant can be defined in a variety of ways, and its numerous applications include the study of unlinking numbers and link concordance. In this survey, we recall the three and four dimensional definitions of $σ_L$, list its main properties and applications, and give comprehensive references for the proofs of these statements.
Linear Independence of Knots Arising from Iterated Infection Without the Use of Tristram Levine Signatures
利用ρ^1-不变量给出了在纽结和谐群的（n）-可解滤子中任意深的线性无关纽结族的显式构造.以前的和谐群中的无限秩子群的构造和我们的构造之间的区别是，我们提出的构造中最深的感染结允许具有消失的Tristram Levine签名。
Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1\times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the classical Levine-Tristram invariant of a knot. In the 3-dimensional situation, a count of singular connections on a knot complement reproduces the Levine-Tristram invariant. We compute the invariant for a number of examples embedded tori, and show that our topological invariant is the same as what one might expect from Echeverria's invariant. Langte Ma has subsequen…
We give a new proof that the Levine-Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Friedl and Teichner to show that the 4-genus of a link does not increase under infection by a string link, which is a generalised satellite construction, provided that certain homotopy triviality conditions hold on the axis curves, and that enough Milnor's invariants of the infection…
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 …
Given a knot $K$ inside an integer homology sphere $Y$, the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into $SU(2)$ which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4-manifold $X$ with the integral homology of $S^{1}\times S^{3}$, and an embedded torus $T$ inside $X$ such that $H_{1}(T;\mathbb{Z})$ surjects onto $H_{1}(X;\…
A Proof of the Kashaev Signature Conjecture
本文利用有向链环及其反向链环不交并上的Seifert曲面定义，证明了Kashaev关于图示链环不变量等于Levine-Tristram符号差两倍的猜想，并给出Alexander多项式的新公式。
We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.
On the Kashaev signature conjecture
本文证明Kashaev关于链环图的签名猜想在经典签名情形下成立，并联系Kauffman模型，从而对任意链环验证了Alexander多项式和经典签名部分，对确定纽结验证了完整猜想。
Generalized Seifert surfaces and signatures of colored links
本文利用“广义塞弗特曲面”将Levine-Tristram签名推广到S^3中的有色链上。这在m维环面上产生一个整数值函数，其中m是链接的颜色数。m=1的情况对应于Levine-Tristram签名。我们发现，后者不变量的许多显着的性质扩展到这个m-变量的推广：它消失的非手性有色链接，它是“分段连续”，和跳跃的地方是由亚历山大不变量的有色链接。使用一个4维的解释和Atiyah-Singer G-签名定理，我们还证明了这个签名是不变的彩色和谐，它提供了一个下界的“切片属”的彩色链接。
On the signatures of torus knots
研究了环面纽结$T_{p,q}$的特征函数的性质。首先，我们提供了一个非常基本的证明公式的积分的签名在圆。我们还得到了环面纽结的Tristram-Levine签名的一个关于Dedekind和的闭公式。
We present a proof of Litherland's formula for the Tristram-Levine signature of a satellite knot in terms of its constituents. Litherland's original proof used more advanced algebraic techniques, while ours uses only linear algebra and some basic results in knot theory.
The average signature of graph links
我们计算平均Tristram-Levine签名的任何图链接的正重量在三个球体，推广Kirby和Melvin的结果。主要工具是计算图链接的等变签名的Neumann算法和Dedekind和的互易定律。
Deformations of singularities of plane curves. Topological approach
我们利用一个纽结不变量，即Tristram-Levine签名，研究了平面曲线的奇点的变形。我们找到了一个界限的M数的总和在所有的奇异性的一个通用的纤维在M数的奇异性在中心纤维和一些拓扑数据。
We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine-Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.
Knot signature functions are independent
对每个具有正虚部的单位复数定义了一个Tristram-Levine纽结签名函数。所有这样的签名函数的集合作为在所有节点的集合上定义的函数的集合是线性无关的。平均签名函数的集合形成结和谐群上的同态的线性无关集合。然而，对于亚历山大多项式的每个单位根，在该根及其共轭处存在具有非零签名的切片结，而在其他地方则没有。这些结果适用于所有奇数维的纽结。
The four-genus of connected sums of torus knots
本文研究了环面纽结的线性组合的四个亏格：aT（p，q）# -bT（p '，q'）.固定正的p、q、p '和q'，我们的重点是四个亏格作为正的a和b的函数的行为。三种类型的例子：在第一，对于所有的a和b的四个属是完全确定的Tristram-莱文签名功能;第二，最近定义的厄普函数的Ozsvath-Stipsicz-Szabo确定所有的a和b的四个属;第三，一个令人惊讶的签名和厄普之间的相互作用出现。
We refine prior bounds on how the multivariable signature and the nullity of a link change under link cobordisms. The formula generalizes a series of results about the 4-genus having their origins in the Murasugi-Tristram inequality, and at the same time extends previously known results about concordance invariance of the signature to a bigger set of allowed variables. Finally, we show that the multivariable signature and nullity are also invariant under $0.5$-solvable cobordism.
A diagrammatic computation of abelian link invariants
本文提出通过单个对称矩阵计算着色链接的多变量签名和alexander多项式，并扩展了kauffman的行列式模型。
Twisted acyclicity of a circle and link signatures
具有非平凡局部系数的圆的同调是平凡的。从这一众所周知的事实，我们推导出几何推论的联系余维2。特别地，Murasugi-Tristram签名被扩展到由任意定向的奇数维球面的闭余维2子流形形成的链接的不变量。新颖之处在于，子流形不被假设为不相交的，但彼此是横向的，并且签名由整个环面的点参数化。Murasugi-Tristram不等式及其推广也被推广到这种情形。
Extended signatures and link concordance
本文研究了扩展Levine-Tristram签名作为连通性不变量，并提出了一个无法被传统方法检测的3链环族。
Signature of rotors
Tutte在图论中引入了转子。1987年，Anstee、Przytycki和Rolfsen将这一概念应用于纽结理论，作为突变的概括。在本文中，我们证明了Tristram-Levine签名是保持方向保持旋转。此外，我们证明了从Goeritz矩阵的特征多项式中得到的任何链接不变量，包括Murasugi签名，都不会因旋转而改变。在2001年，P. Traczyk证明了任何一对保持方向的旋转体的Conway多项式是重合的。但反向旋转是否保持Conway多项式仍是一个悬而未决的问题。我们证明了存在一对具有不同Conway多项式的反向转动子。这提供了对问题的消极解决方案。
Torres-type formulas for link signatures
本文用三维与四维两种方法研究多变量链环签名函数在变量趋于 1 时的极限，给出其向完整环面的扩展及 Torres 型公式，并得到近 1 处 Levine-Tristram 签名的新估计。
A lower bound for the genus of a knot using the Links-Gould invariant
通过研究Links-Gould多项式次数，给出了任意纽结Seifert亏格的下界，改进了Alexander多项式的Seifert不等式，并检测到Kinoshita-Terasaka和Conway纽结的亏格至少为2。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。