An invariant of links in a thickened torus

A polynomial invariant depending on three variables is constructed for links in a thickened torus. The construction involves Kauffman’s formal knot theory based on the Dehn presentation of the knot group. Certain properties of the invariant are established, and a theorem about a Conway type relation is proved. Bibliography: 10 titles.     

Links of surface singularities and CR space forms

We classify 3-dimensional compact locally homogeneous non-degenerate CR-manifolds (“CR space-forms”). Most of them are links of normal complex surface singularities, and we classify these singularities. In memory of Peter Scherk Research partially supported by the NSF.     

Knotoids and knots in the thickened torus

We study the relationship between knotoids and knots in the direct product of the two-dimensional torus and an interval. Each knotoid on the sphere can be lifted to a knot of geometric degree 1 in the thickened torus. We prove that lifting is a bijection on the set of prime knotoids of complexity greater than 1.     

Tabulating knots in the thickened Klein bottle

We tabulate all knots in the oriented thickened Klein bottle having diagrams with three crossings and less. For proving that the knots are distinct, we use a generalization of the Kauffman bracket polynomial in four variables.     

Routeing in a Network with Multi-Class Links

Consider a communication network with certain nodes and different types of links. In addition to the normal link cost, a transformation cost is charged if the incoming link and the outgoing link are of different types. An optimal routeing from a given node to its destination node is sought. The major difficulty in handling this problem is that the principle of optimality does not hold. A model with node separation is built to overcome this difficulty. By using the new model, the original routeing problem is no more than a shortest-path problem. Hence, we can implement this model to current electronic switching machines.     

An invariant of links in the thickened torus

An invariant of links with two and more components in the thickened torus is constructed; the invariant depends on several variables. The construction uses Kauffman’s formal theory, which is based on Dehn’s representation of knot groups. This invariant is a natural generalization of a polynomial z constructed by Zenkina and Manturov. Some properties of the new invariant are also considered.     

Line Graph Links

Acta Mathematicae Applicatae Sinica, English Series - It is well known that a shaded link diagram corresponds to a signed plane multi-graph. In graph theory, line graph is an old and important...     

On Alternating Quasipositive Links

Doklady Mathematics - An effectively verifiable condition for quasipositivity of links is given. In particular, it is proven that if a quasipositive link can be represented by an alternating...     

Links between science and mathematics in school

There exist obvious links between the science and mathematics taught in schools. This paper discusses the responses of trainee teachers in science and mathematics to a number of questions relevant to both subjects and points to failures of understanding and of communication. It further seeks to highlight problems of liaison between subject teachers.     

Diffeomorphic vs Isotopic Links in Lens Spaces

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper, we provide a set of moves on disk, band and grid diagrams that connects diffeo-equivalent links: there are up to four isotopy equivalent links in each diffeo-equivalence class. Moreover, we investigate how the diffeo-equivalence relates to the lift of the link in the 3-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in L(p, q) up to diffeo-equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift.     

Links between subderivatives and subdifferentials

We provide formulas linking the radial subderivative to other subderivatives and subdifferentials for arbitrary extended real-valued lower semicontinuous functions.     

Links Between HX-Groups and Hypergroups

The concept of an HX-group is an upgrade of the concept of a group,in which a new operation is defined on the family of non-empty subsets of a group.If this new support set together with the new operation is a group,then we call it an HX-group.On the other hand,a hyperoperation is a mapping having the same codomain as the operation of an HX-group,i.e.,the family of non-empty subsets of the initial set,but a different domain-the set itself.This could be (and was indeed) a source of confusion,which is clarified in this paper.Moreover,HX-groups naturally lead to constructions of hypergroups.The links between these two algebraic concepts are presented,with the aim of reviving the old notion of an HX-group in the current research on algebraic hyperstructures.One of such existing links and one newly established link are also discussed.     

On Invariants of Virtual Links

We propose a new method of generalizing classical link invariants for the case of virtual links. In particular, we have generalized the knot quandle, the knot fundamental group, the Alexander module, and the coloring invariants. The virtual Alexander module leads to a definition of VA-polynomial that has no analogue in the classical case (i.e. vanishes on classical links).     

Links with splitting number one

Geometriae Dedicata - We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also...     

Chord Diagrams and Coxeter Links

The paper presents a construction of fibered links (K, ) outof chord diagrams L. Let be the incidence graph of L. Undercertain conditions on L the symmetrized Seifert matrix of (K,) equals the bilinear form of the simply-laced Coxeter system(W, S) associated to and the monodromy of (K, ) equals minusthe Coxeter element of (W, S). Lehmer's problem is solved forthe monodromy of these Coxeter links.