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In this paper we work toward the HOMFLYPT skein module of $L(p, 1)$, $\mathcal{S}(L(p,1))$, via braids. Our starting point is the linear Turaev-basis, $\Lambda^{\prime}$, of the HOMFLYPT skein module of the solid torus ST, $\mathcal{S}({\rm ST})$, which can be decomposed as the tensor product of the "positive" ${\Lambda^{\prime}}^+$ and the "negative" ${\Lambda^{\prime}}^-$ sub-modules, and the Lambropoulou invariant, $X$, for knots and links in ST, that captures $S({\rm ST})$. It is a well-known result by now that $\mathcal{S}(L(p, 1))=\frac{\mathcal{S}(ST)}{}$, where bbm's (braid band moves) denotes the isotopy moves that correspond to the surgery description of $L(p, 1)$. Namely, a HOMFLYPT-type invariant for knots and links in ST can be extended to an invariant for knots and links in $L(p, 1)$ by imposing relations coming from the performance of bbm's and solving the infinite system of equations obtained that way. \smallbreak In this paper we work with a new basis of $\mathcal{S}({\rm ST})$, $\Lambda$, and we relate the infinite system of equations obtained by performing bbm's on elements in $\Lambda^+$ to the infinite system of equations obtained by performing bbm's on elements in $\Lambda^-$ via a map $I$. More precisely we prove that the solutions of one system can be derived from the solutions of the other. Our aim is to reduce the complexity of the infinite system one needs to solve in order to compute $\mathcal{S}(L(p,1))$ using the braid technique. Finally, we present a generating set and a potential basis for $\frac{\Lambda^+}{}$ and thus, we obtain a generating set and a potential basis for $\frac{\Lambda^-}{}$. We also discuss further steps needed in order to compute $\mathcal{S}(L(p,1))$ via braids., Comment: 26 pages, 6 figures. arXiv admin note: text overlap with arXiv:1802.09376, arXiv:1702.06290

An ω-limit set of a continuous self-mapping of a compact metric space X is said to be totally periodic if all of its points are periodic. In [3] Askri and Naghmouchi proved that if f is a one-to-one continuous self mapping of a regular curve, then every totally periodic ω-limit set of f is finite. This also holds whenever f is a monotone map of a local dendrite by Abdelli in [1] . In this paper we generalize these results to monotone maps on regular curves. On the other hand, we give some remarks related to expansivity and totally periodic ω-limit sets for every continuous map on compact metric space.

This paper is dedicated to Professor Jinjin Li on the occasion of his 60th anniversary. In this paper, a new cardinal function called the strong Hausdorff number in semitopological groups is introduced. We mainly show that every paratopological group with countable strong Hausdorff number is ω-admissible. Applying this result, we prove that every bounded set in a paratopological group with countable strong Hausdorff number is strongly bounded, which partially answers to a question posed by Sanchez and Tkachenko.

Let P be a property of topological groups. We say that a topological group G is MinAP modulo P if its image f [ G ] under each continuous homomorphism f : G → K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P: finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G / n ( G ) of every topological group G with respect to its von Neumann kernel n ( G ) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group ( G / n ( G ) ) + equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification.

A quasi-continuous dynamical system is a pair $(X,f)$ consisting of a topological space $X$ and a mapping $f: X\to X$ such that $f^n$ is quasi-continuous for all $n \in \mathbb N$, where $\mathbb N$ is the set of non-negative integers. In this paper, we show that under appropriate assumptions, various definitions of the concept of topological transitivity are equivalent in a quasi-continuous dynamical system. Our main results establish the equivalence of topological and point transitivity in a quasi-continuous dynamical system. These extend some classical results on continuous dynamical systems in [3], [10] and [25], and some results on quasi-continuous dynamical systems in [7] and [8]., 14

In this article we investigate which subsets of R n generate the whole R n by the use of positive integers taken as multipliers, as well as a related question of which subsemigroups of R n are generated by different subsets of R n . In particular, we characterize σ-compact subsets of R n which generate the whole R n , and show that the characterization does not work outside this class of sets. Various examples are suggested.

In this paper, we investigate the special Kahler geometry of the base of the Hitchin integrable system in terms of spectral curves. We give explicit formulas of the metric for some special cases by using the affine coordinates introduced by Freed in 1999. We also find that the Taylor expansion of the special Kahler metric about any point in the base may be computed by topological recursion.

In 2018, Funakoshi, Hashizume, Ito, Kobayashi, and Murai used a deformation of spherical curves called deformation type $\alpha$. Then, it was showed that if two spherical curves $P$ and $P'$ are equivalent under the relation consisting of deformations of type RI and type RIII up to ambient isotopy, and satisfy certain conditions, then $P'$ is obtained from $P$ by a finite sequence of deformations of type $\alpha$. In this paper, we introduce a new type of deformations of spherical curves, called deformation of type $\beta$. The main result of this paper is: Two spherical curves $P$ and $P'$ are equivalent under (possibly empty) deformations of type RI and a single deformation of type RIII up to ambient isotopy if and only if reduced(P) and reduced(P') are transformed each other by exactly one deformation which is of type RIII, type $\alpha$, or type $\beta$ up to ambient isotopy, where reduced(Q) is the spherical curve which does not contain a $1$-gon obtained from a spherical curve $Q$ by applying deformations of type RI up to ambient isotopy., Comment: 13 pages, 14 figures

We classify oriented ribbon 2-knots of 1-fusion with length up to six. We show the difference by: the Alexander polynomial; the trace set, which is obtained from the representations of the knot group to SL ( 2 , C ) ; the twisted Alexander polynomial.

In this paper, we characterize stratifiable (or semi-stratifiable) spaces, and monotonically countably paracompact (or monotonically countably metacompact) spaces by expansions of locally upper bounded semi-continuous poset-valued maps. These extend earlier results for real-valued Locally bounded functions., Comment: 15 pages

The notion of cut points on a virtual link diagram was introduced by H. A. Dye and it was extended by the author to the notion of oriented cut points. Using oriented cut points, one can often apply some arguments working only on almost classical virtual link diagrams to any virtual link diagram. In this paper we construct a multivariable polynomial X D for a virtual link diagram using oriented cut points. It turns out that this polynomial is the same with the multivariable polynomial invariant of virtual links introduced by H. A. Dye and L. H. Kauffman and by Y. Miyazawa independently. Using this fact, we give some properties of the multivariable polynomial invariant and some applications.

For coprime integers p ( > 0 ) and q, the ( p , q ) -cable Γ-polynomial of a knot K is the Γ-polynomial of the ( p , q ) -cable knot of K, where the Γ-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we give some results on Vassiliev knot invariants derived from the cable Γ-polynomials. In particular, we show that all Vassiliev knot invariants of order ≤4 are determined by the cable Γ-polynomials.

A generalized torsion element is a non-trivial element such that some non-empty finite product of its conjugates is the identity. We construct a generalized torsion element of the fundamental group of a 3-manifold obtained by Dehn surgery along a knot in the 3-sphere., 15 pages, 9 figures. We fixed the problem in figures

Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in $3$-space. Cocycles of them can be used to construct state-sum type invariants of links and handlebody-links. In this paper we discuss cocycles of a certain class of biquandles and multiple conjugation biquandles, which we call $G$-Alexander biquandles and $G$-Alexander multiple conjugation biquandles, with a relationship with group cocycles. We give a method to obtain a (biquandle or multiple conjugation biquandle) cocycle of them from a group cocycle., Comment: 31 pages

In this paper, we give a lower bound for the maximum and minimum genus of a multibranched surface by the first Betti number and the minimum and maximum genus of the boundary of the neighborhood of it, respectively. As its application, we show that the maximum and minimum genus of $G\times S^1$ is equal to twice of the maximum and minimum genus of $G$ for a graph $G$, respectively. This provides an interplay between graph theory and 3-manifold theory., Comment: 9 pages, 2 figures

In this paper, we shall consider the hyperspace of all nontrivial convergent sequences S c ( X ) of a Frechet-Urysohn nondiscrete space X, which is equipped with the Vietoris topology. We study the spaces X for which S c ( X ) is Baire: this kind of spaces have a dense subset of isolated points (see [12] ). We characterize the spaces X for which S c ( X ) is Baire. This characterization uses a topological game inspired by the Banach-Mazur game. As a consequence, we obtain that if X is completely metrizable and has a dense subset of isolated points, then S c ( X ) is Baire. Our last main result shows that S c ( X ) is pseudocompact iff X is homeomorphic to the one-point compactification of a discrete space of uncountable size. This last assertion provides a characterization of the one-point compactification of a discrete space of uncountable size.

In this paper, we mainly prove that open k-to-one maps need not inversely preserve regular G δ -diagonals (or zero-set diagonals), and open k-to-one maps preserve zero-set diagonals. If X is continuously semi-metrizable, then the Vietoris hyperspace F ( X ) is also continuously semi-metrizable. For a Hausdorff space X, the Pixley-Roy hyperspace F 2 [ X ] has a regular G δ -diagonal if and only if each point of X is a G δ -set.

Recently, Andradi, Shen, Ho and Zhao asked whether can find a complete characterization of T 0 spaces for the SI-convergence being topological. In this paper, we give a positive answer to this problem. More precisely, we introduce the notion of I ⁎ -continuous spaces, and prove that the SI-convergence in a T 0 space is topological if and only if the T 0 space is I ⁎ -continuous.

Each continuous weak selection for a space X defines a coarser topology on X, called a selection topology. Spaces whose topology is determined by a collection of such selection topologies are called continuous weak selection spaces. For such spaces, Garcia-Ferreira, Miyazaki, Nogura and Tomita considered the minimal number cws ( X ) of selection topologies which generate the original topology of X, and called it the cws-number of X. In this paper, we show that cws ( X ) ≤ 2 for every semi-orderable space X, and that cws ( X ) = 2 precisely when such a space X has two components and is not orderable. Complementary to this result, we also show that cws ( X ) = 1 for each suborderable metrizable space X which has at least 3 components.

In this article, we introduce two notions which are called property (σ-A) and property (σ-B). They are generalizations of property (A) and property (B), respectively. Every space with a point-countable base (or σ-NSR pair-base) satisfies property (σ-A). Every space with the Collins-Roscoe property satisfies property (σ-B). We show that every compact Hausdorff space with property (σ-A) is metrizable. Thus some known conclusions can be generalized. This shows that property (σ-A) plays a key role in the metrizability of compact Hausdorff spaces. We show that the properties of property (σ-A) and property (σ-B) are closed under finite products. Every finite product of T 1 -spaces which satisfy property (σ-B) (property (σ-A), σ-sheltering (F), σ-well-ordered (F)) is hereditarily a D-space. If ( X , T ) satisfies ω 1 -sheltering (F), then ( X , T ω ) is hereditarily a D-space. We show that if a space X satisfies ω 1 -sheltering (F) and every countable discrete subspace of X is closed, then X is hereditarily a D-space. This gives a partial answer to a question posed by Z.Q. Feng and J.E. Porter in 2015. We finally give examples to show that there exists a space which has property (σ-A) but it does not have a point-countable base and there exists a space which has property (C) but it does not have property (σ-A).

Connectedness (resp. uniform connectedness [14] ) of uniform spaces can be defined in terms of continuous (resp. uniformly continuous) functions to a discrete space requiring that every continuous (resp. uniformly continuous) function to a discrete space has to be constant. Replacing uniformly continuous functions by the strictly weaker notion of Cauchy regular functions [15] we obtain a new notion of connectedness, namely Cauchy connectedness which happens to be an intermediate notion between connectedness and uniform connectedness. We primarily investigate several features of this new notion. Further in metric spaces, we turn our attention to quasi-Cauchy sequences [10] and show that replacing Cauchy regular continuity by ward continuity (functions preserving quasi-Cauchy sequences, see [11] ) one again gets back the notion of uniform connectedness. Finally we define WC spaces in line of UC spaces and obtain certain characterizations.

Two families of one-cone torus bundles parametrized by cone angles are studied. When the cone angles are sufficiently small, the combinatorial structures of the Dirichlet domains with respect to the basepoints in the singular locus are characterized.

For a finite tree T and its endpoint v, we can define a natural partial order ≤ such that v is the greatest element in T. For a compact metric space X and a continuous map f : X → T , let ↓ f = { ( x , t ) : t ≤ f ( x ) } and ↓ C ( X , T ) = { ↓ f : f is continuous from X to T } . We investigate the subspace ↓ C ( X , T ) of the family of all non-empty closed sets in X × T with the Hausdorff distance. The following result is proved: For every non-degenerate Peano continuum X and a tree T with n segments { S 1 , S 2 , ⋯ , S n } , ↓ C ( X , T ) ≈ ⨁ i = 1 n ↓ C U B ( S i ) , where C U B ( S i ) = { f ∈ C ( X , T ) : max ⁡ f ( X ) ∈ S i } . Moreover, every ↓ C U B ( S i ) is contractible space. As a conclusion, ↓ C ( X , T ) has n connected component.

Let X : = [ M / G ] be a compact orbifold given by the quotient of a connected compact Lie group G action on an oriented smooth closed manifold M, where the action is assumed to be smooth, effective, locally free and orientation preserving. Let f ¯ : X → R be a Morse function which lifts to a G-invariant Morse–Bott function f on M. We construct an explicit isomorphism between the Morse–Bott cohomology of f and the orbifold Morse cohomology of f ¯ by constructing a new filtered cochain complex out of basic forms to connect the Morse–Bott cochain complex of f and the orbifold Morse–Smale–Witten cochain complex of f ¯ .

A continuum X has unique cone provided that the following property holds: if Y is a continuum and c o n e ( X ) is homeomorphic to c o n e ( Y ) , then X is homeomorphic to Y. A fan is an arcwise continuum with exactly one ramification point having the property that the intersection of every two subcontinua is connected. This paper is devoted to prove that a fan has unique cone if and only if it is not the cone over a zero-dimensional non-degenerate compact metric space.

A multiple conjugation quandle is an algebra whose axioms are motivated from handlebody-knot theory. Any linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair. In this paper, we show that any affine extension of a multiple conjugation quandle can be described by using a quadruple of maps, called an augmented MCQ Alexander pair., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1905.02519

In this paper, the concepts of g s 2 -convergence and quasi-liminf convergence of filters in posets are introduced. The main results are: (1) For an order consistent topology τ on a poset P, P is τ-quasicontinuous iff the g s 2 -convergence coincides with τ-convergence; (2) For an order consistent topology τ on a poset P, the quasi-liminf convergence coincides with τ ∨ ω ( P ) -convergence if P is τ-quasicontinuous; (3) For an order consistent topology τ on a poset P, P is τ-continuous iff the s 2 -convergence coincides with τ-convergence iff it is meet τ-continuous and the quasi-liminf convergence coincides with τ ∨ ω ( P ) -convergence.

In this paper, we define near-compact mappings and characterize the sequence-covering, near-compact images of metric spaces. The following main result is obtained. The following are equivalent for a space X: (1) X is a 1-sequence-covering near-compact image of a metric space. (2) X is a sequence-covering near-compact image of a metric space. (3) X has a sequence of cs-covers which is a point-finite point-star network at N S ( X ) . (4) X has a sequence of sn-covers which is a point-finite point-star network at N S ( X ) .

In this paper we establish connections between the theory of Katětov order on ideals on countable sets with ideal convergence in general topological spaces, which are used to study the following questions posed by X.G. Zhou, L. Liu and S. Lin [12] : (1) Whether every finite union of I -closed subsets is I -closed? (2) Must every I -continuous map preserve I -convergence? (3) Must every map preserving I -convergence preserve J -convergence if J ⊆ I ? Our main results include: • If I is K-uniform, then every finite union of I -closed subsets is I -closed in any space; • there exists a countable zero-dimensional Hausdorff space with character equal to the continuum in which there are two I -closed sets with non- I -closed union for some tall F σ -ideal I , while • it is independent of ZFC that for every Hausdorff space X of character less than the continuum and every tall F σ -ideal I , finite unions of I -closed subsets are always I -closed in X. These answers Question (1). We show the answer to Question (2) is negative in ZFC and the answer to Question (3) is also negative if we replace J ⊆ I with J ≤ K I (which is weaker than J ⊆ I ).

We say that a (para)topological group G is strongly submetrizable if it admits a coarser separable metrizable (para)topological group topology and is projectively strongly submetrizable if for each open neighborhood U of the identity in G, there is a closed invariant subgroup N contained in U such that the quotient (para)topological group G / N is strongly submetrizable. We show that a quotient group of a simply sm-factorizable ω-narrow topological abelian group can fail to be simply sm-factorizable. This answers a question posed by Arhangel'skii and the first listed author in 2018. If, however, the kernel of a quotient homomorphism is a bounded subgroup, then the homomorphism preserves simple sm-factorizability in the classes of topological and paratopological groups. We also prove that a regular (para)topological group G is simply sm-factorizable if and only if G is projectively strongly submetrizable and every continuous real-valued function on G is uniformly continuous on G ω , the P-modification of G. Making use of this fact we show that all weakly Lindelof projectively strongly submetrizable paratopological groups and all weakly Lindelof paratopological abelian groups are simply sm-factorizable. It is also established that every precompact paratopological group is simply sm-factorizable.

For a based CW-complex X, A ♯ n ( X ) is the submonoid of [ X , X ] which consists of all homotopy classes of self-maps of X that induce an automorphism on π k ( X ) for all 0 ≤ k ≤ n . Since, for m n , A ♯ n ( X ) ⊆ A ♯ m ( X ) , there is a chain by inclusions: E ( X ) ⊆ A ♯ ∞ ( X ) ⊆ . . . ⊆ A ♯ 1 ( X ) ⊆ A ♯ 0 ( X ) = [ X , X ] . In this paper, we study the number of strict inclusions in this chain for a given connected CW-complex. We call this number the self-length of a given space. We prove that the self-length is a homotopy invariant and investigate the close connection with the self-closeness number, which is the minimum number n such that E ( X ) = A ♯ n ( X ) . Moreover, we determine self-lengths of several spaces and provide the lower bounds or upper bounds of the self-lengths of some spaces.

It is know that the Valdivia compact spaces can be characterized by a special family of retractions called r-skeleton (see [10] ). Also we know that there are compact spaces with r-skeletons which are not Valdivia. In this paper, we shall study r-skeletons and special families of closed subsets of compact spaces. We prove that if X is a zero-dimensional compact space and { r s : s ∈ Γ } is an r-skeleton on X such that | r s ( X ) | ≤ ω for all s ∈ Γ , then X has a dense subset consisting of isolated points. Also we give conditions to an r-skeleton in order that this r-skeleton can be extended to an r-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a Σ-product of a power of the unit interval. Following this fact we introduce the notion of π-skeleton on a compact space X by embedding X in a suitable power of the unit interval together with a pair ( F , φ ) , where F is family of metric separable subspaces of X and φ an ω-monotone function which satisfy certain properties. This new notion generalize the idea of a Σ-product. We prove that a compact space admits a r-skeleton iff it admits a π-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with r-skeletons admits an r-skeleton (see [9] ).