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In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

This paper is a continuation of [2]. Working in the context of commutative differential graded algebras, we study the ideal of the cohomology classes which can be annihilated by fibrations whose fiber has finite homological dimension. In this paper we identify these classes with certain higher products in cohomology. [ABSTRACT FROM AUTHOR]

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]

A many-to-many k -disjoint path cover (k -DPC) of a graph G is a set of k vertex-disjoint paths joining k distinct pairs of source and sink in which each vertex of G is contained exactly once in a path. The balanced hypercube B H n , a variant of the hypercube, was introduced as a desired interconnection network topology. Let S = { s 1 , s 2 , ... , s 2 n − 2 } and T = { t 1 , t 2 , ... , t 2 n − 2 } be any two sets of vertices in different partite sets of B H n (n ≥ 2). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of B H n when | S | = | T | = 2. In this paper, we prove that there exists unpaired many-to-many (2 n − 2) -disjoint path cover of B H n (n ≥ 2) from S to T , which has improved some known results. The upper bound 2 n − 2 is best possible in terms of the number of disjoint paths in unpaired many-to-many k -DPC of B H n . [ABSTRACT FROM AUTHOR]

An -string tangle is a pair such that is a disjoint union of properly embedded arcs in a topological -ball . And an -string tangle is said to be trivial (or rational), if it is homeomorphic to as a pair, where is a 2-disk, is the unit interval and each is a point in the interior of . A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an -string stick tangle its stick-order is defined to be a nonincreasing sequence of natural numbers such that, under an ordering of the arcs of the tangle, each denotes the number of sticks constituting the th arc of the tangle. And a stick-order is said to be trivial, if every stick tangle of the order is trivial. In this paper, restricting the -ball to be the standard 3-ball, we give the complete list of trivial stick-orders. [ABSTRACT FROM AUTHOR]

Bigeneralized topology was introduced in 2010. In this paper, we aim to contribute in the development of this area by proving results with respect to the hereditary class that defines two types of Lindelöf spaces in a bigeneralized topological space (BGTS). Some results on strong BGTS (briefly SBGTS) are proved by defining a -perfect mapping. We also prove some equivalent conditions with respect to the hereditary class. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the research results of DAD, Daily Artificial Dispatcher, published in 2010 [S. Z. Stefanov, New Mathematics and Natural Computation6 (2010) 275–283], and give it new interpretation bring out the best of its meaning, as well as some more quantitative novel formula. In metaphorical sense, DAD is the analogue to DNA and hence the Smart Grid is analogue to the Creature by invoking the postmodern theory [J.-F. Lyotard, Moralités Postmodernes (Éditions Galilée, 1993) (in French)]. Specifically, the DAD’s binary expressions are generated by an innocent dialogue between DAD and Smart Grid in the form of world-strings. Namely, the living space-time of the DAD’s binary symbols is generated via a discussion between DAD and Smart Grid. Metaphorically speaking, DAD’s world is digitally described as a dramatic game and the Smart Grid as a creating cartoon loaded with an integral holographic complexity. Overall, the so called creatures capable of innocent discussions under near-zero temperature are generated by the epidemic growth of the Smart Grid cartoon. It is further concluded that the number of the Smart Grid creatures is inversely proportional to the half-time of the DAD’s binary “DNA” life cycle. Each Smart Grid can be coded in one of eight colors pool, as an aging in one of the four ways and as being in one of three possible development phases, dynamically. Finally, we take the liberty of calling the DAD’s world, a string prescribed as topology and landscape, to be “Aria”. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and investigate some separation axioms in bigeneralized topological spaces. Using the concepts of regular generalized star b-open sets due to Indirani and Sindhu, the study defines and characterizes -, -, -regular and -normal spaces. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss spacetime with space-matter tensor and some properties related to it. Finally, we construct an example of a mixed quasi-Einstein spacetime. [ABSTRACT FROM AUTHOR]

We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of L κ + , κ + -indescribability and Π ξ 1 -indescribability of a cardinal κ for all ξ < κ + . In this context, universal Π ξ 1 formulas exist, there is a normal ideal associated to Π ξ 1 -indescribability and the notions of Π ξ 1 -indescribability yield a strict hierarchy below a subtle cardinal. Additionally, given a regular cardinal μ , we introduce a diagonal version of Cantor's derivative operator and use it to extend Bagaria's [Derived topologies on ordinals and stationary reflection, Trans. Amer. Math. Soc. 371(3) (2019) 1981–2002] sequence 〈 τ ξ : ξ < μ 〉 of derived topologies on μ to 〈 τ ξ : ξ < μ + 〉. Finally, we prove that for all ξ < μ + , if there is a stationary set of α < μ that have a high enough degree of indescribability, then there are stationarily many α < μ that are nonisolated points in the space (μ , τ ξ + 1). [ABSTRACT FROM AUTHOR]

Since topoi were introduced, there have been efforts putting mathematics into the context of topoi. Amongst known topoi, the topoi of sheaves or presheaves over a small category are of special interest. We have here as the base topos that of sheaves over a monoid M as a one object category. By means of closure operators we then obtain categories of sheaves related to the right ideals of M. These categories have already been studied but we give these categories a more thorough treatment and reveal some additional properties. Namely, for a weak topology determined by a right ideal I of M , we show that the category of sheaves associated to this topology is a subtopos of M S e t (the presheaves over M) and determine the Lawvere–Tierney topology yielding the same subtopos, which is the Lawvere–Tierney topology associated to the idempotent hull of the (not necessarily idempotent) closure operator associated to I. We will then find conditions under which the subcategory of separated objects turns out to be a topos, and in the last section, we find conditions under which the category of sheaves becomes a De Morgan topos. [ABSTRACT FROM AUTHOR]

Let (X ,) be a topological space. By the Skula topology (or the b -topology) on X , we mean the topology b () on X with basis the collection of all -locally closed sets of X , the resulting space (X , b ()) will be denoted by b (X). We show that the following results hold: (1) b (X) is an Alexandroff space if and only if the T 0 -reflection T 0 (X) of X is a T D -space. (2) b (X) is a Noetherian space if and only if T 0 (X) is finite. (3) If we denote by X * the Alexandroff extension of X , then b (X *) = (b (X)) * if and only if (X ,) is a Noetherian quasisober space. We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, b (b (b (X))) = b (b (X)). A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology Clop () of : The indiscrete components of Clop (X) are of the form C x = { x } ¯ ∩ (x) , where x ∈ X and (x) is the intersection of all open sets of X containing x (equivalently, C x = { y ∈ X : { x } ¯ = { y } ¯ }). We show that Clop (X) = b (b (X)) [ABSTRACT FROM AUTHOR]

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it. [ABSTRACT FROM AUTHOR]

We study knots of order in the grope filtration and the solvable filtration of the knot concordance group. We show that, for any integer , there are knots generating a subgroup of . Considering the solvable filtration, our knots generate a subgroup of distinct from the subgroup generated by the previously known -torsion knots of Cochran, Harvey, and Leidy. We also present a result on the -torsion part in the Cochran, Harvey, and Leidy's primary decomposition of the solvable filtration. [ABSTRACT FROM AUTHOR]