Recent zbMATH articles in MSC 57https://zbmath.org/atom/cc/572021-01-08T12:24:00+00:00WerkzeugCharacter varieties of even classical pretzel knots.https://zbmath.org/1449.570022021-01-08T12:24:00+00:00"Chen, Haimiao"https://zbmath.org/authors/?q=ai:chen.haimiaoThe purpose of the present paper under review is to determine the character variety of irreducible \(\text{SL}(2,\mathbb{C})\)-representations for even \(3\)-strand pretzel knots. The variety is explicitly calculated, and it turns out to have 4 components, two of which have dimension 0 and the others of which have dimension 1. This completes the calculation for \(3\)-strand pretzel knots with the author's previous work, \textit{H. Chen} [Int. J. Math. 29, No. 9, Article ID 1850060, 15 p. (2018; Zbl 1432.57010)]. The last part discusses a method of computing the \(A\)-polynomials for even \(3\)-strand pretzel knots.
Reviewer: Masakazu Teragaito (Hiroshima)Counting surface branched covers.https://zbmath.org/1449.570042021-01-08T12:24:00+00:00"Petronio, Carlo"https://zbmath.org/authors/?q=ai:petronio.carlo"Sarti, Filippo"https://zbmath.org/authors/?q=ai:sarti.filippoTo a branched cover \(f\) between orientable surfaces one can associate a certain branch datum \(\mathcal{D}(f)\), that encodes the combinatorics of the cover. This \(\mathcal{D}(f)\) satisfies a compatibility condition called the Riemann-Hurwitz relation. Let \(f:\tilde{\Sigma}\rightarrow \Sigma\) be a surface branched cover of some degree \(d\). If there are \(n\) branching points and we order them in some arbitrary fashion, the local degrees at the preimages of the \(j\)-th point form a partition \(\pi_j\) of \(d\), and we define \(\mathcal{D}(f)=(\tilde{\Sigma}, \Sigma, d, n, \pi_1,\ldots,\pi_n)\).
If \(\pi_j\) has length \(l_j\), the Riemann-Hurwitz relation states: \[\chi(\tilde{\Sigma})-(l_1+\ldots+l_n)=d (\chi(\Sigma)-n).\] The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum \(\mathcal{D}\) there exists a branched cover \(f\) such that \(\mathcal{D}(f)= D\). In this paper the authors restrict \(\Sigma\) to be orientable and ask how many of these branched covers \(f\) exist under a suitable equivalence relation. They show that quite a few natural choices for this relation are possible and give a careful analysis that shows that the number of actually distinct ones is only three. Finally, they show that these three are indeed different.
Reviewer: Claus Ernst (Bowling Green)Topological loops with six-dimensional solvable multiplication groups having five-dimensional nilradical.https://zbmath.org/1449.220032021-01-08T12:24:00+00:00"Figula, Ágota"https://zbmath.org/authors/?q=ai:figula.agota"Ficzere, Kornélia"https://zbmath.org/authors/?q=ai:ficzere.kornelia"Al-Abayechi, Ameer"https://zbmath.org/authors/?q=ai:al-abayechi.ameerSummary: Using connected transversals we determine the six-dimensional indecomposable solvable Lie groups with five-dimensional nilradical and their subgroups which are the multiplication groups and the inner mapping groups of three-dimensional connected simply connected topological loops. Together with this result we obtain that every six-dimensional indecomposable solvable Lie group which is the multiplication group of a three-dimensional topological loop has one-dimensional centre and two- or three-dimensional commutator subgroup.Study on constructions and properties of tangent module bundles.https://zbmath.org/1449.550032021-01-08T12:24:00+00:00"Zhang, Feijun"https://zbmath.org/authors/?q=ai:zhang.feijunSummary: In this paper, the tangent module bundles are constructed on a manifold. We give a smooth structure on the tangent module bundles which makes it a smooth manifold. Then we discuss the properties of the tangent module bundles and give the bracket product of Poisson on the module bundles such that it is a Lie algebra.Trace-free \(SL(2,\mathbb{C})\)-representations of arborescent links.https://zbmath.org/1449.570012021-01-08T12:24:00+00:00"Chen, Haimiao"https://zbmath.org/authors/?q=ai:chen.haimiaoThe paper extends results about trace-free representations given in \textit{H. Chen} [J. Knot Theory Ramifications 27, No. 8, Article ID 1850050, 10 p. (2018; Zbl 1401.57008)] to the larger class of arborescent links. Given a link \(L\in S^3\), a representation \(\pi_1(S^3-L) \rightarrow SL(2,\mathbb{C})\) is trace-free if it sends each meridian to an element with trace zero. Trace free representations are of interest since they carry information on the geometry of link complements, for example each trace-free \(SL(2,\mathbb{C})\)-representation of a link \(L\) gives rise to a representation \(\pi_1(M_2(L)) \rightarrow SL(2,\mathbb{C})\), where \(M_2(L)\) is the double covering of \(S^3\) branched along \(L\). The paper presents a method for completely determining trace-free \(SL(2,\mathbb{C})\)-representations for arborescent links. The method uses trace-free representations of arborescent tangles \(T\) (these are two-string tangles that can contain circular components of the link) defined as a homomorphism \(\pi_1(B^3-T) \rightarrow SL(2,\mathbb{C})\) sending each meridian to an element of \(SL^0(2,\mathbb{C})\). The method relies on the build-up of arborescent tangles, starting with rational tangles via iterated horizontal and vertical composition of tangles. In the last section concrete computations are done for a class of 3-bridge arborescent links.
Reviewer: Claus Ernst (Bowling Green)On equilibrium triangulations of quasitoric manifolds.https://zbmath.org/1449.570072021-01-08T12:24:00+00:00"Datta, Basudeb"https://zbmath.org/authors/?q=ai:datta.basudeb"Sarkar, Soumen"https://zbmath.org/authors/?q=ai:sarkar.soumenSummary: Quasitoric manifolds introduced by previous researchers are topological generalizations of smooth complex projective spaces. In 1992, a 10-vertex equilibrium triangulation of \(\mathbb{CP}^2\) was constructed. In this paper, we generalize this construction for quasitoric manifolds and provide an algorithm for constructing equilibrium triangulation of several quasitoric manifolds. In some cases our construction gives vertex minimal equilibrium triangulations.Generalized Han-Liu-Zhang's anomaly cancellation formulas.https://zbmath.org/1449.580022021-01-08T12:24:00+00:00"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.5|wang.yong.7|wang.yong.10|wang.yong.3|wang.yong.8|wang.yong.6|wang.yong.1|wang.yong|wang.yong.9|wang.yong.2"Wu, Tong"https://zbmath.org/authors/?q=ai:wu.tongSummary: In [in: Frontiers in differential geometry, partial differential equations and mathematical physics. In memory of Gu Chaohao. Hackensack, NJ: World Scientific. 87--104 (2014; Zbl 1299.81039)], \textit{F. Han} et al. gave a anomaly cancellation formula which generalized the Green-Schwarz formula and the Schwartz-Witten formula. In this paper, we give two generalized Han-Liu-Zhang formulas and also a Han-Liu-Zhang formula in odd dimension. By studying modular invariance properties of some characteristic forms, some new anomaly cancellation formulas in odd dimension are given.The \(n\)-Gordian complex of knots.https://zbmath.org/1449.570032021-01-08T12:24:00+00:00"Zhang, Kai"https://zbmath.org/authors/?q=ai:zhang.kai"Yang, Zhiqing"https://zbmath.org/authors/?q=ai:yang.zhiqingSummary: In this paper, the \(n\)-Gordian complex of knots is defined which is an extension of the Gordian complex of knots. The authors show that if \(n \ge 2\) and \(n\) is even, then for any knot \(K\) and any positive integer \(r\), there is a finite set of knots of size \(r\) containing \(K\), such that the \(n\)-Gordian distance between any two knots in this set is 1.Involutory Latin quandles of order \(pq\).https://zbmath.org/1449.200662021-01-08T12:24:00+00:00"Jedlička, Přemysl"https://zbmath.org/authors/?q=ai:jedlicka.premyslSummary: We present a construction of a family of involutory latin quandles, a family that contains all non-Alexander involutory latin quandles of order \(pq\), for \(p < q\) odd primes. Such quandles exist if and only if \(p\) divides \(q^2-1\).Classifying spaces for projections of immersions with controlled singularities.https://zbmath.org/1449.570092021-01-08T12:24:00+00:00"Szűcs, A."https://zbmath.org/authors/?q=ai:szucs.andras"Terpai, T."https://zbmath.org/authors/?q=ai:terpai.tamasThe authors present explicit descriptions of classifying spaces and related fibrations for certain maps with special properties. Applications to the geometric study of singularities are also given.
Reviewer: Dumitru Motreanu (Perpignan)Simplicial volume with \(\mathbb{F}_p\)-coefficients.https://zbmath.org/1449.570082021-01-08T12:24:00+00:00"Löh, Clara"https://zbmath.org/authors/?q=ai:loh.claraThe classical simplicial volume \(||\cdot||_\mathbb{R}\) of an oriented closed connected manifold, introduced by \textit{M. Gromov} [Publ. Math., Inst. Hautes Étud. Sci. 56, 5--99 (1982; Zbl 0516.53046)], is defined in terms of the \(l^1\)-norm on the singular chain complex with real coefficients. Ignoring weights the author studies a simplicial volume \(|| \cdot ||_{(R)}\) for any subring \(R \subseteq \mathbb{R}\). This leads to an \(\mathbb{F}_p\)-version \(||\cdot||_{(\mathbb{F}_p)}\). The author discusses relations between such volumes (varying coefficients) and Betti numbers. The following questions are addressed:
What is the topological/geometric meaning of these invariants?
How do these invariants relate to other topological invariants?
(How) does the choice of the prime \(p\) effect the simplicial volume?
Do these invariants relate to the weightless \(\mathbb{Q}\)-simplicial volume?
What happens if \(\mathbb{F}_p\)-simplicial volumes are stabilized along a tower of finite coverings?
Reviewer: Karl Heinz Dovermann (Honolulu)Invariant elements in the dual Steenrod algebra.https://zbmath.org/1449.550042021-01-08T12:24:00+00:00"Vergili, T."https://zbmath.org/authors/?q=ai:vergili.tane"Karaca, I."https://zbmath.org/authors/?q=ai:karaca.ismetSummary: In this paper, we investigate the invariant elements of the dual mod \(p\) Steenrod subalgebra \(\mathcal{A}_p^*\) under the conjugation map \(\chi\) and give bounds on the dimensions of \((\chi-1)(\mathcal{A}_p^*)_d\), where \((\mathcal{A}_p^*)_d\) is the dimension of \(\mathcal{A}_p^*\) in degree \(d\).Coloring properties of a class of oblique and diagonal symmetry tangles.https://zbmath.org/1449.570052021-01-08T12:24:00+00:00"Wang, Shuxin"https://zbmath.org/authors/?q=ai:wang.shuxin"Wang, Dongxue"https://zbmath.org/authors/?q=ai:wang.dongxue"Li, Siyu"https://zbmath.org/authors/?q=ai:li.siyu"Wang, Hetong"https://zbmath.org/authors/?q=ai:wang.hetongSummary: This paper discusses and gives the color matrices of a class of oblique and diagonal symmetry tangles by the coloring rules of tangles. On this basis, it gives the color matrices of mirror image of tangles, horizontal flip tangles and vertical flip tangles corresponding to the class of oblique and diagonal symmetry tangles.Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4.https://zbmath.org/1449.050762021-01-08T12:24:00+00:00"Fulek, Radoslav"https://zbmath.org/authors/?q=ai:fulek.radoslav"Kynčl, Jan"https://zbmath.org/authors/?q=ai:kyncl.janThe strong Hanani-Tutte theorem [\textit{W. T. Tutte}, J. Comb. Theory 8, 45--53 (1970; Zbl 0187.20803)] states that a graph is planar if it can be drawn in the plane such that no pair of independent edges cross an odd number of times.
This paper provides a counterexample to the possible extension of the strong Hanani-Tutte theorem for orientable surfaces of genus 4. Namely, the authors construct a graph of genus 5 that has a drawing on the orientable surface of genus 4, with every pair of independent edges crossing an even number of times.
Moreover, they construct a counterexample to the possible extension of the unified Hanani-Tutte theorem [\textit{R. Fulek} et al., Electron. J. Comb. 24, No. 3, Research Paper P3.18, 8 p. (2017; Zbl 1369.05045)] on the torus.
Reviewer: Stelian Mihalas (Timişoara)The cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces.https://zbmath.org/1449.570102021-01-08T12:24:00+00:00"Zhao, Xu'an"https://zbmath.org/authors/?q=ai:zhao.xuan|zhao.xu-anSummary: In this paper we introduce the history and current status of the computation for the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces. We give some problems worthy of further attention and research.On the cup-length of certain classes of flag manifolds.https://zbmath.org/1449.570062021-01-08T12:24:00+00:00"Balko, L."https://zbmath.org/authors/?q=ai:balko.ludovit"Lörinc, J."https://zbmath.org/authors/?q=ai:lorinc.jurajThe \(\mathbb{Z}_2\)-cup-length of a topological space \(X\) is the largest number of elements in the cohomology ring of \(X\), with coefficients in \(\mathbb{Z}_2\), whose product is non-trivial (this gives a lower bound for the Ljusternik-Schnirelmann category of \(X\)). The elements of the flag manifold \(F(n_1, \ldots, n_q)\), for positive integers \(n_i\) whose sum is \(n\), are the sets of all flags of type \((n_1, \ldots, n_q)\) in \(\mathbb{R}^n\), i.e. of all sets of \(q\) mutually orthogonal subspaces of \(\mathbb{R}^n\) of dimensions \((n_1, \ldots, n_q)\) (which can be identified with the homogeneous space \(O(n)/O(n_1) \times \cdots \times O(n_q)\)); a special case are the Grassmann manifolds \(F(k,n)\).
In the present paper, the cup-length of the flag manifolds \(F(2,2,n_3)\) and \(F(1,3,2^{s+1}-3)\) is computed; similar computations for various flag manifolds are contained in papers by \textit{J. Korbaš} and \textit{J. Lörinc} [Fundam. Math. 178, No. 2, 143--158 (2003; Zbl 1052.55006)] and by \textit{Z. Z. Petrović} et al. [Acta Math. Hung. 149, No. 2, 448--461 (2016; Zbl 1389.57006)]. In the second part of the present paper the authors compute the height (the largest nontrivial power) of the third Stiefel-Whitney characteristic class of the canonical vector bundle over the Grassmann manifold \(F(4,n)\); analogous results for the first and second Stiefel-Whitney classes are contained in papers by \textit{R. E. Stong} [Topology Appl. 13, 103--113 (1982; Zbl 0469.55005)] and by \textit{S. Dutta} and \textit{S. S. Khare} [J. Indian Math. Soc., New Ser. 69, No. 1--4, 237--251 (2002; Zbl 1104.57301)].
Reviewer: Bruno Zimmermann (Trieste)