Search

The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows: (1) The annulus sum H = H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either H1 or H2; (2) Let H = H1∪Sg,b H2 be a surface sum of two handlebodies H1 and H2 along a connected surface S = Sg,b,,b ≥ 1, ni = g(Hi) ≥ 2, i = 1, 2. Suppose that S is incompressible in both H1 and H2. Then H is a handlebody if and only if there exists a basis J = {J}1,…, Jm with a partition (J1, J2) of J such that J1 is primitive in H1 and J2 is primitive in H2.

We introduce the [Formula: see text]-splittings for 3-manifolds as follows. For a compact connected surface [Formula: see text] properly embedded in a compact connected orientable 3-manifold [Formula: see text], if [Formula: see text] decomposes [Formula: see text] into two handlebodies [Formula: see text] and [Formula: see text], then [Formula: see text] is called an [Formula: see text]-splitting for [Formula: see text]. Clearly, when [Formula: see text] is closed, this is just the Heegaard splitting for [Formula: see text]; when [Formula: see text] is with boundary, the [Formula: see text]-splitting for [Formula: see text] is different from the Heegaard splitting for [Formula: see text]. In this paper, we first show that any compact connected orientable 3-manifold admits an [Formula: see text]-splitting, then generalize Casson–Gordon theorem on weakly reducible Heegaard splitting to the [Formula: see text]-splitting case in the following version: if [Formula: see text] is a weakly reducible [Formula: see text]-splitting for a compact connected orientable 3-manifold [Formula: see text], then (1) [Formula: see text] contains an incompressible closed surface of positive genus or (2) the [Formula: see text]-splitting [Formula: see text] is reducible or (3) there is an essential 2-sphere [Formula: see text] in [Formula: see text] such that [Formula: see text] is a collection of essential disks in [Formula: see text] and [Formula: see text] is an incompressible and not boundary parallel planar surface in [Formula: see text] with at least two boundary components, where [Formula: see text] or (4) [Formula: see text] is stabilized.

In this paper, we weaken the nice condition of an n-dimensional rendezvous task defined in the work of X. Liu et al [11] . Then we introduce the definition of evolution of non-degenerate n-dimensional rendezvous task. A non-degenerate n-rendezvous task is said to be evolution if the q-th reduced homology group of its decision space is abelian group for q = n and trivial for the others. Well-known examples are set agreement, simplex agreement, and approximation agreement and so on. Each n-rendezvous task is assigned an algebraic signature, which consists of n-th homology group of the decision space, as well as a distinguished element in the group. We show that an evolution of non-degenerate n-dimensional rendezvous task implements another if and only if there is a homomorphism from its signature to the other. Hence the computational power of evolution of non-degenerate rendezvous task is completely characterized by its signature. Last, we talk about the degenerate n-dimensional rendezvous task in which the output values in any execution can construct at most an n-dimensional simplex.

Приводятся конструкция и обзор свойств полинома Ямады пространственных графов, и даются формулы полиномов Ямады для некоторых классов графов. Строится бесконечное семейство пространственных графов, для которых корни полиномов Ямады образуют плотное подмножество комплексной плоскости.

This article combines the principal component analysis (PCA) with persistent homology for applications in biomolecular data analysis. We extend the technique of persistent homology to localized weighted persistent homology to fit the properties of molecules. We introduce this novel PCA in the study of the folding process of residues 1 to 28 of amyloid beta peptide in solution. We are able to determine seven metastable states of amyloid beta 1 to 28 using homology of dimension 2, corresponding to seven local minimums in the free energy landscape. We also give the transition information between the seven types and the disconnectivity graph. Our result is very robust under change of parameters. Furthermore persistent homology of dimension 1 also give consistent results. This method can be applied to different peptides and molecules.

In a 21st century context characterized by the inefficient use of urban construction land, to optimize land use structure and transform resource utilization methods, it is necessary for the Chinese government to improve land use efficiency. Promoting the readjustment of inefficient urban land use has become essential. The purpose of this paper is to sort through the policies addressing the readjustment of inefficient urban land use in China, identify the promulgation date, longitudinal hierarchy, and horizontal composition of the relevant policies, and summarize the evolutionary trend of such policies. This study aims to provide a reference for the adjustment and improvement of relevant policy systems in China. The content analysis method was used in this study, and NVivo 12 software was used to compare and verify the analysis process. The results show the following: (1) A large number of relevant policies have been promulgated, with a total of 12 items from 1988 to 2012, increasing to six items annually on average from 2013 to 2019. The Communist Party of China Central Committee, the National People&rsquo, s Congress of the People&rsquo, s Republic, the State Council and relevant ministries promulgated 13.46%, 11.54%, 28.85%, and 46.15%, respectively, of the relevant policies. (2) The existing policies are mostly issued in the form of notices, opinions, management regulations, etc., therefore, their level of effectiveness is low, and the role of norms is weak. (3) Finally, the policy content experienced stages of initial exploration, basic establishment, and continuous improvement. It also covered aspects such as Standard Control, Market Configuration, Overall Planning, Incentive Mechanisms, Income Distribution, and Supervision Evaluation. The system for policies addressing readjustment of inefficient urban land use has basically been perfected, although post approval supervision still needs to be strengthened. Therefore, the process of readjusting inefficient urban land use should be optimized, new modes of readjustment should be constantly explored, and inefficient land use should be addressed across the entire territory.

Suppose V ∪SW is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F1 and F2 are pairwise disjoint homeomorphic essential subsurfaces in ∂_V. In this paper, we give a suffcient condition such that the self-amalgamation of V ∪SW along F1 and F2 is unstabilized and uncritical.

A Heegaard splitting V 1 ∪ S V 2 is called weakly keen with distance 2 if there are essential separating disks D i ⊂ V i such that a component of V i − D i is homeomorphic to F i × I and there is an unique geodesic { ∂ D 1 , C 0 , ∂ D 2 } in C ( S ) connecting ∂ D 1 to ∂ D 2 , where C 0 is an essential simple closed curve in S and F i is a component of ∂ − V i for i = 1 , 2 . In this paper, we give a sufficient condition for the weakly keen Heegaard splittings to be keen. At last, we give a sufficient condition for the self-amalgamation of a weakly keen Heegaard splitting to be unstabilized.

Let M be a compact connected 3-submanifold of the 3-sphere S3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S1, · · ·, S n } properly embedded in M, ∂S = {∂S1, · · ·, ∂S n } is a complete curve system on F. We call S a complete surface system for M, and ∂S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ∂S. As an application of the result, it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup H n .

The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra, which is an approach similar to Jones’s original construction.

We survey the construction and properties of the Yamada polynomial of spatial graphs and present the Yamada polynomial formulae for some classes of graphs. Then we construct an infinite family of spatial graphs for which roots of Yamada polynomials are dense in the complex plane., 15 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:1801.09075

In this paper we give a complicated distributional chaos, that is, a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set.

Let [math] be the link group of a Brunnian [math] –link [math] and [math] be the normal closure of the [math] meridian in [math] for [math] . In this article, we show that the intersecting subgroup [math] coincides with the iterated symmetric commutator subgroup [math] for [math] using the techniques of homotopy theory. Moreover, we give a presentation for the intersecting subgroup [math] .

We study a Khovanov type homology close to the original Khovanov homology theory from Frobenius system. The homology is an invariant for oriented links up to isotopy by applying a tautological functor on the geometric complex. The homology has also geometric descriptions by introducing the genus generating operations. We prove that Jones Polynomial is equal to a suitable Euler characteristic of the homology groups. As an application, we compute the homology groups of (2, k)-torus knots for every k ∈ N.

A link tower is a sequence of links with the structure given by removing the last components. Given a link tower, we prove that there is a chain complex consisting of (non-abelian) groups given by the symmetric commutator subgroup of the normal closures in the link group of the meridians excluding the meridian of the last component with the differential induced by removing the last component. Moreover, the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis. These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.

In the paper, we give an equivalent description of the lens space [Formula: see text] with [Formula: see text] prime in terms of any corresponding Heegaard diagrams as follows: Let [Formula: see text] be a closed orientable 3-manifold with [Formula: see text] and [Formula: see text] a Heegaard splitting of genus [Formula: see text] for [Formula: see text] with an associated Heegaard diagram [Formula: see text]. Assume [Formula: see text] is a prime integer. Then [Formula: see text] is homeomorphic to the lens space [Formula: see text] if and only if there exists an embedding [Formula: see text] such that [Formula: see text] bounds a complete system of surfaces for [Formula: see text].

In this paper, we will give a sufficient condition for self-amalgamation of a strongly irreducible Heegaard splitting along any essential subsurface to be unstabilized.

Let V ∪ S W be a Heegaard splitting of M, such that ∂ M = ∂_W = F 1 ∪ F 2 and g(S) = 2g(F 1) = 2g(F 2). Let V* ∪ S* W* be the self-amalgamation of V ∪ S W. We show if d(S) ⩾ 3 then S* is not a topologically minimal surface.

There are three versions of distributional chaos, namely DC1, DC2 and DC3. By using an example of constant-length substitution system, we show that DC3 need not imply Li–Yorke chaos. (In this paper, chaos means the existence of an uncountable scrambled set of the corresponding type, while the existing example only deals with a single pair of points.)

It is known that in a compact dynamical system, the whole space can be a Li–Yorke scrambled set, but this does not hold for distributional chaos. In this paper we prove that the complement of a distributionally scrambled set must be an infinite set. Then we give an example of an uncountable dense invariant open extremal distributionally scrambled set which is the complement of a countable infinite set. This presents one kind of the “largest” (from the topological point of view) distributionally scrambled set in a compact dynamical system. Moreover, we construct an uncountable closed distributionally scrambled set.

For a closed orientable surface Sg of genus not smaller than 2, C(Sg) is the curve complex on Sg whose vertices consist of the isotopy classes of nontrivial circles on Sg. It has been showed that any two vertices in C(Sg) can be connected by an edge path, and C(Sg) has an infinite diameter. We show that for 0 ⩽ i ⩽ 3g−5, two i-simplices can be connected by an (i +1)-path in C(Sg), and the diameter of C(Sg) under such a distance is infinite.

Let [Formula: see text] be a compact metric space and [Formula: see text] be two continuous maps on [Formula: see text]. The iterated function system [Formula: see text] is the action of the semi-group generated by [Formula: see text] on [Formula: see text]. In this paper, we introduce the definitions of shadowing property, average shadowing property and topological ergodicity for [Formula: see text] and give some examples. Then we show that (1) if [Formula: see text] has the shadowing property then so do [Formula: see text] and [Formula: see text]; (2) [Formula: see text] has the shadowing property if and only if the step skew product corresponding to [Formula: see text] has the shadowing property. At last, we prove a Lyapunov stable iterated function system having the average shadowing property is topologically ergodic.

In this paper, we investigate the simplicial groups obtained from the link groups of naive cablings on any given framed link. Our main result states that the resulting simplicial groups have the homotopy type of the loop space of a wedge of 3 3 -spheres. This gives simplicial group models for some loop spaces using link groups.

This paper is concerned with the crosscap numbers of Montesinos knots. For a family of Montesinos knots, we give a lower bound of [Formula: see text] among all essential surfaces [Formula: see text] embedded in the exterior, where [Formula: see text] denotes the ratio of negative Euler characteristic of the surface and the number of sheets. From this result, we determine the crosscap numbers of a family of Montesinos knots. Our method relies on the algorithm of enumerating all essential surfaces for Montesinos knots given by Hatcher and Oertel.

In the present paper, we show that the Morimoto Conjecture on the super additivity of the tunnel numbers of knots in S 3 S^3 is true for knots K 1 , K 2 K_1,K_2 in S 3 S^3 in which each K i K_i is either a tunnel number one or m m -small, i = 1 , 2 i=1,2 . This extends two known results by Morimoto.

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m, then surgery on some ( m − 1 ) components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.

Let V∪SW be a reducible Heegaard splitting of genus g = g(S) ⩾ 2. For a maximal prime connected sum decomposition of V ∪SW, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the decomposition, and p the number of all other prime factors in the decomposition. The main result of the present paper is to describe the relation of p, q and dim(CV ∩ CW).

It is known that a maximal collection of essential annuli in H2 could contain exactly 1, or 2, or at most 3 annuli, and it is clear how they are embedded in H2. It is also known that for a maximal collection [Formula: see text] of essential annuli in Hn with n ≥ 3, [Formula: see text], and the bounds are sharp. Moreover, for each integer m, 2 ≤ m ≤ 4n - 5, there exists a maximal collection of essential annuli in Hn which contains exactly m annuli. In this paper, we detect how a maximal collection of essential annuli in Hn with n ≥ 3 are embedded in Hn.

In this paper, we show the following result: Let K i be a knot in a closed orientable 3-manifold M i such that (M i ,K i ) is not homeomorphic to (S 2 ×S 1, x 0 ×S 1), i = 1, 2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E(K i ) is less than the difference of one and twice of the tunnel number of K i . Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive. We further show that if the distance of a Heegaard splitting of each knot complement is strictly bigger than twice the tunnel number of the knot (twice the sum of the tunnel number of the knot and one, respectively), then the tunnel number of connected sum of two such knots is additive (super additive, respectively).

In this article, we investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups., 33 pages

Let [math] be a Heegaard splitting for a closed orientable [math] –manifold [math] . The inclusion-induced homomorphisms [math] and [math] are both surjective. The paper is principally concerned with the kernels [math] , [math] , their intersection [math] and the quotient [math] . The module [math] is of special interest because it is isomorphic to the second homotopy module [math] . There are two main results. ¶ (1) We present an exact sequence of [math] –modules of the form ¶ ( K ∩ L ) ∕ [ K , L ] ↪ R { x 1 , … , x g } ∕ J → T ϕ R { y 1 , … , y g } → θ R ↠ ϵ ℤ , ¶ where [math] , [math] is a cyclic [math] –submodule of [math] , [math] and [math] are explicitly described morphisms of [math] –modules and [math] involves Fox derivatives related to the gluing data of the Heegaard splitting [math] . ¶ (2) Let [math] be the intersection kernel for a Heegaard splitting of a connected sum, and [math] , [math] the intersection kernels of the two summands. We show that there is a surjection [math] onto the free product with kernel being normally generated by a single geometrically described element.

Let M be a compact orientable closed 3-manifold, and F a non-separating incompressible closed surface in M. Let M ' = M (F), where (F) is an open regular neighborhood of F in M. In the paper, we give a lower bound of genus of self-amalgamation of minimal Heegaard splitting V ' ( SW ' of M ' under some conditions on the distance of the Heegaard splitting.

Let [Formula: see text] be a dynamical system, [Formula: see text] be the system on hyperspace and [Formula: see text] be its [Formula: see text]-fuzzification. We study the relations among the various forms of sensitivity of the systems [Formula: see text], [Formula: see text] and [Formula: see text]. Firstly, we give a simpler proof of the theorem in [Wu & Chen, 2017] which states that [Formula: see text] is sensitive if and only if [Formula: see text] is sensitive. Then, we show that (1) all forms of sensitivity of [Formula: see text] are the same for [Formula: see text], (2) all forms of sensitivity of [Formula: see text] partly imply the same for [Formula: see text], and the converse holds in some cases.

The main purpose of this paper is to further explore the complexity of fuzzified dynamical systems. Especially, we study several kinds of specification properties of Zadeh’s extension. Among other things, we discuss the “stronger” sensitivity on product dynamical systems of g-fuzzification. There are two major ingredients. Firstly, it is proved that the specification (respectively almost specification) property of the original system and its Zadeh’s extension is equivalent, when the original system has the shadowing property. Moreover, we study the [Formula: see text]-sensitivity (respectively multi-sensitivity) of g-fuzzification and its induced product dynamical system.

We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labeled graphs is obtained.

Suppose [Formula: see text] is a Heegaard splitting and [Formula: see text] is an essential separating disk in [Formula: see text] such that a component of [Formula: see text] is homeomorphic to [Formula: see text], [Formula: see text]. In this paper, we prove that if there is a locally complicated simplicial path in [Formula: see text] connecting [Formula: see text] to [Formula: see text], then the geodesic connecting [Formula: see text] to [Formula: see text] is unique. Moreover, we give a sufficient condition such that [Formula: see text] is keen and the geodesic between any pair of essential disks on the opposite sides has local uniqueness property.

Let Mi be a connected, compact, orientable 3-manifold, Fi a boundary component of Mi with g(Fi) ⩾ 2, i = 1, 2, and F1 ≊ F2. Let φ: F1 → F2 be a homeomorphism, and M = M1 ∪φM2, F = F2 = φ(F1). Then it is known that g(M) ⩽ g(M1)+g(M2)−g(F). In the present paper, we give a sufficient condition for the genus of an amalgamated 3-manifold not to go down as follows: Suppose that there is no essential surface with boundary (Qi, ∂Qi) in (Mi, Fi) satisfying χ(Qi) > 3 − 2g(Mi), i = 1, 2. Then g(M) = g(M1) + g(M2) − g(F).