Recurrent Generalization of F-Polynomials for Virtual Knots and Links

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.

Linking numbers, quandles and groups

We introduce a quandle invariant of classical and virtual links, denoted by [Formula: see text]. This quandle has the property that [Formula: see text] if and only if the components of [Formula: see text] and [Formula: see text] can be indexed in such a way that [Formula: see text], [Formula: see text] and for each index [Formula: see text], there is a multiplier [Formula: see text] that connects virtual linking numbers over [Formula: see text] in [Formula: see text] to virtual linking numbers over [Formula: see text] in [Formula: see text]: [Formula: see text] for all [Formula: see text]. We also extend to virtual links a classical theorem of Chen, which relates linking numbers to the nilpotent quotient [Formula: see text].

Virtual and arrow Temperley–Lieb algebras, Markov traces, and virtual link invariants

Let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and standard polynomials in the variables [Formula: see text] For [Formula: see text] we denote by [Formula: see text] the virtual braid group on [Formula: see text] strands. We define two towers of algebras [Formula: see text] and [Formula: see text] in terms of diagrams. For each [Formula: see text] we determine presentations for both, [Formula: see text] and [Formula: see text]. We determine sequences of homomorphisms [Formula: see text] and [Formula: see text], we determine Markov traces [Formula: see text] and [Formula: see text], and we show that the invariants for virtual links obtained from these Markov traces are the [Formula: see text]-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each [Formula: see text] the standard Temperley–Lieb algebra [Formula: see text] embeds into both, [Formula: see text] and [Formula: see text], and that the restrictions to [Formula: see text] of the two Markov traces coincide.

On arrow polynomials of checkerboard colorable virtual links

In this paper, we give two new criteria of detecting the checkerboard colorability of virtual links by using the odd writhe and the arrow polynomial of virtual links, respectively. As a result, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

Span of the Jones polynomials of certain v-adequate virtual links

It is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.

A low-delay information sharing algorithm for multiple-radio-per-platform networking

In order to ensure the strong real-time information sharing of Aerial Ad hoc Network, a low-delay information sharing algorithm for multiple-radio-per-platform networking is proposed based on the directional transmission capability of phased-array antenna. The algorithm introduces virtual nodes and virtual links in the process of topology generation first. By extracting topology information and choosing link grouping, it can effectively reduce redundant transmission and transmission latency of information sharing. Then, it is verified through simulation that the algorithm can reduce the latency by up to 49.8% and eliminate transmission redundancy. In addition, a direction selection algorithm is proposed for the variation of antenna beam direction. Simulation results show that the algorithm can further reduce the latency of information sharing and ensure the real time of information sharing, thus further improving the network performance.