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An Extension of the (mathfrak {sl}_2) Weight System to Graphs with (n le 8) Vertices (mathfrak的一个扩展{sl}_2)具有(nle 8)顶点的图的权重系统
Q3 Mathematics Pub Date : 2021-09-06 DOI: 10.1007/s40598-021-00187-7
Evgeny Krasilnikov

Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that the value of the weight system constructed from the Lie algebra (mathfrak {sl}_2) on a chord diagram depends on the intersection graph of the diagram, rather than the diagram itself. This inspired a question due to Lando about whether it is possible to extend the weight system (mathfrak {sl}_2) to a graph invariant satisfying the four term relations for graphs. We show that for all graphs with up to 8 vertices such an extension exists and is unique, thus answering in affirmative to Lando’s question for small graphs.

弦图和四项关系是瓦西里耶夫在1980年末提出的。权重系统的各种构造是已知的,并且每种这样的构造都产生结不变量。特别地,权重系统可以由李代数以及所谓的图的4变量构造。Chmutov–Lando定理指出,由李代数构造的权重系统的值{sl}_2)弦图取决于图的交集图,而不是图本身。这激发了Lando提出的一个问题,即是否有可能扩展重量系统{sl}_2)到满足图的四项关系的图不变量。我们证明了对于所有具有8个顶点的图,这样的扩展是存在的并且是唯一的,从而肯定地回答了Lando关于小图的问题。
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引用次数: 2
Element-Building Games on (mathbb {Z}_n) 元素构建游戏在(mathbb{Z}_n)
Q3 Mathematics Pub Date : 2021-08-18 DOI: 10.1007/s40598-021-00185-9
Bret Benesh, Robert Campbell

We consider a pair of games where two players alternately select previously unselected elements of (mathbb {Z}_n) given a particular starting element. On each turn, the player either adds or multiplies the element they selected to the result of the previous turn. In one game, the first player wins if the final result is 0; in the other, the second player wins if the final result is 0. We determine which player has the winning strategy for both games except for the latter game with nonzero starting element when (n in {2p,4p}) for some odd prime p.

我们考虑一对游戏,其中两个玩家交替选择先前未选择的(mathbb)元素{Z}_n)给定特定的起始元素。在每个回合中,玩家将他们选择的元素与上一回合的结果相加或相乘。在一场比赛中,如果最终结果为0,则第一名选手获胜;在另一种情况下,如果最终结果为0,则第二个玩家获胜。当(nin{2p,4p })为某个奇数素数p时,我们确定除了后一个具有非零起始元素的博弈之外,哪一个博弈对两个博弈都有获胜策略。
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引用次数: 0
New Invariants of Poncelet–Jacobi Bicentric Polygons Poncelet–Jacobi双中心多边形的新不变量
Q3 Mathematics Pub Date : 2021-08-18 DOI: 10.1007/s40598-021-00188-6
Pedro Roitman, Ronaldo Garcia, Dan Reznik

The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville’s theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family’s limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the (N=4) case).

介于两个圆之间的庞塞莱多边形的1d族被称为双心族。利用椭圆函数和Liouville定理,我们证明了(i)该族具有内角余弦的不变和,以及(ii)关于该族的极限点的踏板多边形具有不变周长。有趣的是,(i)和(ii)都是椭圆台球N周期性的性质。此外,由于(ii)中的踏板多边形与椭圆台球N周期性关于焦点中心圆的逆相同,因此一个重要的推论是(iii)椭圆台球焦点逆N边具有恒定的周长。有趣的是,这些也保留了它们的余弦和(除了(N=4)的情况)。
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引用次数: 8
Cutpoints of Invariant Subcontinua of Polynomial Julia Sets 多项式Julia集不变次连续线的截点
Q3 Mathematics Pub Date : 2021-08-16 DOI: 10.1007/s40598-021-00186-8
Alexander Blokh, Lex Oversteegen, Vladlen Timorin

We prove fixed point results for branched covering maps f of the plane. For complex polynomials P with Julia set (J_{P}) these imply that periodic cutpoints of some invariant subcontinua of (J_{P}) are also cutpoints of (J_{P}). We deduce that, under certain assumptions on invariant subcontinua Q of (J_{P}), every Riemann ray to Q landing at a periodic repelling/parabolic point (xin Q) is isotopic to a Riemann ray to (J_{P}) relative to Q.

我们证明了平面的分支覆盖映射f的不动点结果。对于具有Julia集的复多项式P(J_。我们推导出,在对(J_{P})的不变子连续性Q的某些假设下,每一条到Q的黎曼射线都落在周期排斥/抛物点。
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引用次数: 3
Surfaces of Section for Seifert Fibrations Seifert纤维截面表面
Q3 Mathematics Pub Date : 2021-08-05 DOI: 10.1007/s40598-021-00184-w
Bernhard Albach, Hansjörg Geiges

We classify global surfaces of section for flows on 3-manifolds defining Seifert fibrations. We discuss branched coverings—one way or the other—between surfaces of section for the Hopf flow and those for any other Seifert fibration of the 3-sphere, and we relate these surfaces of section to algebraic curves in weighted complex projective planes.

我们对定义Seifert纤维的3-流形上流动的截面的全局曲面进行了分类。我们讨论了Hopf流的截面曲面和任何其他3-球面的Seifert fibration的截面曲面之间的分支覆盖,并将这些截面曲面与加权复投影平面中的代数曲线联系起来。
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引用次数: 2
On a Theorem of Lyapunov–Poincaré in Higher Dimensions 关于高维Lyapunov–Poincaré的一个定理
Q3 Mathematics Pub Date : 2021-07-13 DOI: 10.1007/s40598-021-00183-x
V. León, B. Scárdua

The classical Lyapunov–Poincaré center theorem assures the existence of a first integral for an analytic 1-form near a center singularity in dimension two, provided that the first jet of the 1-form is nondegenerate. The basic point is the existence of an analytic first integral for the given 1-form. In this paper, we consider generalizations for two main frameworks: (1) real analytic foliations of codimension one in higher dimension and (2) singular holomorphic foliations in dimension two. All this is related to the problem of finding criteria assuring the existence of analytic first integrals for a given codimension one germ with a suitable first jet. Our approach consists in giving an interpretation of the center theorem in terms of holomorphic foliations and, following an idea of Moussu, apply the holomorphic foliations arsenal to obtain the required first integral. As a consequence we are able to revisit some of Reeb’s classical results on integrable perturbations of exact homogeneous 1-forms, and prove versions of these in the framework of non-isolated (perturbations of transversely Morse type) singularities.

经典的Lyapunov–Poincaré中心定理保证了在二维中心奇异点附近的解析1-形式的第一积分的存在,前提是1-形式的首次喷流是非退化的。基本点是对于给定的1-形式存在解析第一积分。在本文中,我们考虑了两个主要框架的推广:(1)高维余维一的实解析叶理和(2)维二的奇异全纯叶理。所有这一切都与寻找一个准则的问题有关,该准则确保了一个给定余维的具有合适的第一射流的一个芽的解析第一积分的存在。我们的方法包括用全纯叶理来解释中心定理,并遵循Moussu的思想,应用全纯叶理库来获得所需的第一积分。因此,我们能够重新审视Reeb关于精确齐次1-形式的可积扰动的一些经典结果,并在非孤立(横向Morse型扰动)奇点的框架下证明这些结果的版本。
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引用次数: 1
Real Lines on Random Cubic Surfaces 随机三次曲面上的实线
Q3 Mathematics Pub Date : 2021-07-02 DOI: 10.1007/s40598-021-00182-y
Rida Ait El Manssour, Mara Belotti, Chiara Meroni

We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface (Zsubset {mathbb {R}}{mathrm {P}}^3) defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group O(4) by change of variables. Such invariant distributions are completely described by one parameter (lambda in [0,1]) and as a function of this parameter the expected number of real lines equals:

$$begin{aligned} E_lambda =frac{9(8lambda ^2+(1-lambda )^2)}{2lambda ^2+(1-lambda )^2}left( frac{2lambda ^2}{8lambda ^2+(1-lambda )^2}-frac{1}{3}+frac{2}{3}sqrt{frac{8lambda ^2+(1-lambda )^2}{20lambda ^2+(1-lambda )^2}}right) . end{aligned}$$

This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to (lambda =frac{1}{3}) and for which (E_{frac{1}{3}}=6sqrt{2}-3.) Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case (lambda =1) and for which (E_1=24sqrt{frac{2}{5}}-3).

我们给出了一个关于随机不变三次曲面上实线数期望的显式公式,即由随机高斯多项式定义的曲面(Zsubet{mathbb{R}}{math rm{P}}}^3),其概率分布在正交群O(4)的作用下通过变量的变化而不变。这种不变分布完全由一个参数(lambdain[0,1])来描述,并且作为该参数的函数,期望的实数等于:$$beagin{aligned}E_lambda=frac{9(8lambda^2+(1-lambda)^2)}{2lambda^2+1-lambda)^2}{20lambda ^2+(1-lambda)^2}right)。end{aligned}$$这个结果推广了Basu等人以前的结果。(Math Ann 374(3-4):1773–181019)对于Kostlan多项式的情况,该多项式对应于(λ=frac{1}{3}),并且(E_{2}-3.)此外,我们证明了实线数的期望通过随机纯谐波三次多项式最大化,这对应于情况(λ=1),并且对于情况(E_1=24sqrt{frac{2}{5}}-3)。
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引用次数: 3
Strange Duality Between the Quadrangle Complete Intersection Singularities 四边形完全交奇异性之间的奇异对偶性
Q3 Mathematics Pub Date : 2021-06-22 DOI: 10.1007/s40598-021-00181-z
Wolfgang Ebeling, Atsushi Takahashi

There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and Wall. We derive this duality from a variation of the Berglund–Hübsch transposition of invertible polynomials introduced in our previous work about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.

在第一作者发现的四边形孤立完全交奇点和Wall之间存在着一种奇怪的对偶性。我们从我们之前的工作中引入的可逆多项式的Berglund–Hübsch转置的变体中导出了这种对偶性,该工作使用大小为2的矩阵因子分解来研究超曲面和完全相交奇点之间的奇异对偶性。
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引用次数: 0
A Galois–Dynamics Correspondence for Unicritical Polynomials 单临界多项式的Galois–动力学对应
Q3 Mathematics Pub Date : 2021-06-09 DOI: 10.1007/s40598-021-00179-7
Robin Zhang

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell–Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton–Patel, and Vivaldi–Hatjispyros in the early 1990s connected the irreducibility and Galois-theoretic properties of dynatomic polynomials to rational periodic points; from the Galois–dynamics correspondence, we derive similar consequences for quadratic periodic points of unicritical polynomials. This is sufficient to deduce the non-existence of quadratic periodic points of quadratic polynomials with exact period 5 and 6, outside of a specified finite set from Morton and Krumm’s work in explicit Hilbert irreducibility.

在与Mordell–Lang猜想证明中使用的阿贝尔变体的扭点的Galois同态性质的类比中,我们描述了Galois群的作用和有理映射的动力学作用之间的对应关系。对于具有有理系数的非线性多项式,相关的动态原子多项式的不可约性是一个方便的标准,尽管我们也验证了当动态原子多项式可约时,在几种情况下会出现对应关系。Morton、Morton–Patel和Vivaldi–Hatjispyros在20世纪90年代初的工作将动态原子多项式的不可约性和伽罗瓦理论性质与有理周期点联系起来;从伽罗瓦-动力学对应关系中,我们导出了单临界多项式的二次周期点的类似结果。这足以从Morton和Krumm在显式Hilbert不可约性中的工作中推导出在指定的有限集之外,具有精确周期5和6的二次多项式的二次周期点的不存在性。
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引用次数: 1
Conjectures on Stably Newton Degenerate Singularities 关于稳定牛顿退化奇点的猜想
Q3 Mathematics Pub Date : 2021-06-07 DOI: 10.1007/s40598-021-00178-8
Jan Stevens

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.

We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, (x^p+x^q) in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

我们讨论了Arnold的一个问题,即每个函数是否稳定地等价于其牛顿图的非退化函数。我们认为答案是否定的。我们描述了一种使函数在稳定后不退化的方法,并给出了该方法不起作用的奇点的例子。我们猜想它们实际上是稳定退化的,即不稳定等价于非退化函数。我们回顾了文献中的各种非退化概念。对于有限特征,我们猜想非退化奇点不存在野生消失环。这意味着具有有限Milnor数的奇点的最简单例子,特征p中的(x^p+x^q),不稳定地等价于非退化函数。我们认为,具有任意数量的Puiseux对(特征为零)的不可约平面曲线是稳定的非退化的。由于稳定涉及许多变量,通常很难确定牛顿图,但方程的形式表明定义函数是非退化的。
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引用次数: 2
期刊
Arnold Mathematical Journal
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