Virtual Links Research Articles

A class of index polynomial invariants for virtual knots using flat virtual knots invariants

In this paper, a class of indices is constructed using a polynomial invariant [Formula: see text] for flat virtual knots. For any integer [Formula: see text] not less than [Formula: see text], the classical crossings of an oriented virtual knot diagram are classified into two types according to whether the intersection index of the classical crossings can be divided by [Formula: see text]. We assign an index to each classical crossing whose intersection index is not divisible by [Formula: see text], the index is related to the flat virtual knot diagram corresponding to the virtual knot diagram obtained by applying 0-smoothing at that classical crossing. We construct a class of index polynomials by using these indices and prove that they are invariants of virtual knots, and also discuss some of their properties.

Intersection graph and writhe polynomial

We prove that two virtual knots have equivalent intersection graphs if and only if they have the same writhe polynomial.

Efficient and secure service function chain deployment method for delay optimization in air traffic information network

The air traffic information network is fundamental for ensuring the efficient and safe operation of flights. However, with the increasing demands of air traffic services, the rigidity of the traditional air traffic information network has become increasingly pronounced. Network function virtualization (NFV) technology presents an effective solution to address this issue. Within the NFV environment, the deployment of service function chains (SFC) forms the basis for achieving end-to-end air traffic services. This paper addresses the low delay and high security requirements inherent in air traffic services and proposes an efficient and secure SFC deployment for delay optimization (ESSFCD-DO) method. Primarily, the paper conducts a theoretical analysis of SFC deployment, and derives four key principles related to SFC delay, security, and resource cost. Subsequently, based on these principles, the ESSFCD-DO method is developed, which encompasses the virtual network function (VNF) aggregation algorithm for security and resources cost optimization (VNFAA-SRCO), the VNF deployment algorithm based on topology aware (VNFD-TA), and the algorithm for deploying virtual links based on k-shortest path. Experimental results demonstrate that, in comparison to other SFC deployment methods within this domain, ESSFCD-DO achieves significant optimizations in terms of end-to-end delay of SFC and long-term revenue-cost ratio, while ensuring security and service request acceptance rate.

Virtualized Delta moves for virtual knots and links

We introduce a local deformation called the virtualized [Formula: see text]-move for virtual knots and links. We prove that the virtualized [Formula: see text]-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient condition for two virtual links to be related by a finite sequence of virtualized [Formula: see text]-moves.

On Knots with Cyclic Symmetries

This review paper investigates the relationship between symmetries of knots and links in the three-dimensional sphere, with a focus on cyclic symmetries, and their associated polynomial invariants. It examines the behavior of these polynomials in the case where the knot is set-wise fixed by the action of finite cyclic group on the 3-sphere. The study highlights how these invariants can obstruct the existence of such symmetries, providing valuable insights into the topological properties of these knots. The paper reviews established results on polynomial criteria for periodicity and their extensions to freely periodic links. It also explores generalizations to study the symmetries of virtual knots and spatial graphs.

MC-biquandles and MC-biquandle coloring quivers

We introduce the notion of mc-biquandles, algebraic structures which have possibly distinct biquandle operations at single-component and multi-component crossings. These structures provide computable homset invariants for classical and virtual links. We categorify these homsets to obtain mc-biquandle coloring quivers and define several new link invariants via decategorification from these invariant quivers.

Optimizing Virtual Network Embedding for Avionics with Time-Triggered Traffic Scheduling

Network virtualization technology abstracts the nodes and links resources in the physical network, and enables multiple virtual networks (VNs) to share the substrate network (SN) resources through the Virtual Netw ork Embedding (VNE) method. For time-triggered Ethernet (TTE) used in avionics, a time-triggered traffic scheduling oriented virtual network embedding (TT-VNE) method is proposed. While meeting the total resource constraints of traditional VNE problem, it ensures the strict periodicity of time-triggered (TT) traffic. During the solution process, the method sorts the virtual nodes according to the TT traffic bandwidth requirements and the total bandwidth requirements of the links connected to the virtual nodes, embeds the virtual nodes using the breadth first search algorithm, and plans the virtual links in candidate shortest path aggregation. If the TT traffic in the current virtual link is not schedulable, the iterative design of local virtual node re embedding and path planning is carried out. The simulation shows that the request acceptance rate of TT-VNE is not lower than that of existing methods, and when the number of virtual network requests in the star topology exceeds 30 , its request acceptance rate is about 14.3%higher than the VNE-NTANRC-D method which only considers the network topology and resource attributes.

Connected sum and crossing numbers of flat virtual knots

Connected sum and crossing numbers of flat virtual knots

Non-abelian tensor product and circular orderability of groups

For a group G we consider its tensor square G⊗G and exterior square G∧G. We prove that for a circularly orderable group G, under some assumptions on H1(G) and H2(G), its exterior square and tensor square are left-orderable. This yields an obstruction for a circularly orderable group G to have torsion. We apply these results to study circular orderability of tabulated virtual knot groups.

A Framework for 5G Network Slicing Optimization using 2-Edge-Connected Subgraphs for Path Protection

Emerging telecommunications technologies require robust frameworks for efficient network slicing. We propose a network-slicing model that aims to optimize the deployment of virtual networks on a physical network topology. Our model ensures compliance with 5G requirements, incorporating latency and capacity constraints on virtual links. Selecting slices with cost and resource requirements on the computing nodes is optimized using a Knapsack problem with revenue maximization. We propose a path protection algorithm to deal with link failures by constructing a 2-edge-connected subgraph (or two link-disjoint Steiner trees) for each slice to provide both primary and backup paths. Simulation results include comparison with existing solutions by metrics such as latency, revenue, resource utilization, number of protected slices, and computation time, providing valuable insights for network planners operating in diverse and dynamic environments. Key contributions include efficient resource allocation using the Knapsack problem, enhanced network resilience via 2-edge-connected subgraphs for path protection, and realistic simulation experiments on SNDlib dataset topologies. The simulation results show that the proposed framework improves computational efficiency compared to the recent related solutions, particularly in large network topologies where k-connected function slicing (KC- FS) subgraph embeddings take approximately 3.5 times more computation time.

Computation of tight bounds for the worst-case end-to-end delay on Avionics Full-Duplex Switched Ethernet

Avionics Full-Duplex Switched Ethernet (AFDX) is a fault-tolerant real-time communication bus for safety–critical applications in aircraft. AFDX configures communication channels, denoted as virtual links (VLs), ensuring bounded message delays through traffic shaping at both end-systems and switches. Effective AFDX network design necessitates computing the worst-case end-to-end delay of time-critical VLs to meet specified message deadlines. This paper presents a new method for calculating tight bounds on the worst-case end-to-end delay for each VL in an AFDX network. We introduce the new notion of an extended uninterrupted transmission interval, which is the prerequisite for computing the worst-case queuing delay at switches. Adding up these queuing delays along the path of each VL between end-systems yields a tight upper bound on the worst-case end-to-end delay. The correctness of our results is formally proved, and comprehensive simulation experiments on different example networks confirm the tightness of our bound. These simulations also demonstrate the superior performance of our method compared to existing approaches that offer more pessimistic as well as optimistic results.

Multivariate Alexander quandles, V. Constructing the medial quandle of a link

In this paper, we explain how the medial quandle of a classical or virtual link can be built from the peripheral structure of the reduced Alexander module.

Clasp pass moves and arrow polynomials of virtual knots

For classical knots, clasp pass moves are closely related to Vassiliev invariants of degree [Formula: see text]. Tsukamoto showed that the values of the Vassiliev invariant of degree [Formula: see text] induced from the Jones polynomial for two knots differ by [Formula: see text] or [Formula: see text], if they are related by a single clasp pass move. For virtual knots, the arrow polynomial is a generalization of the Jones polynomial and induces a Vassiliev invariant of degree [Formula: see text]. We show that the values of the Vassiliev invariant of degree [Formula: see text] induced from the arrow polynomial of two virtual knots differ by [Formula: see text] or [Formula: see text], if they are related by a single clasp pass move. We also obtain a lower bound of the distance between virtual knots by clasp pass moves.

A bracket polynomial for virtual links

In this paper, we describe a method of making an invariant of virtual links, generalizing Jones–Kauffman polynomial of virtual links.

On invariants of multiplexed virtual links

For a virtual knot [Formula: see text] and an integer [Formula: see text] with [Formula: see text], we introduce a method of constructing an [Formula: see text]-component virtual link [Formula: see text], which we call the [Formula: see text]-multiplexing of [Formula: see text]. Every invariant of [Formula: see text] is an invariant of [Formula: see text]. We give a way of calculating three kinds of invariants of [Formula: see text] using invariants of [Formula: see text]. As an application of our method, we also show that Manturov’s virtual [Formula: see text]-colorings for [Formula: see text] can be interpreted as certain classical [Formula: see text]-colorings for [Formula: see text].

Core groups

The core group of a classical link was introduced independently by Kelly in 1991 and Wada in 1992. It is a link invariant defined by a presentation involving the arcs and crossings of a diagram, related to Wirtinger’s presentation of the fundamental group of a link complement. Two close relatives of the core group are defined by presentations involving regions rather than arcs; one of them is related to Dehn’s presentation of a link group. The definitions are extended to virtual link diagrams and properties of the resulting invariants are discussed.

Kinetostatic Modeling of an Asymmetrical Double-Stepped Beam for Displacement Amplification

Abstract A kinetostatic model of an asymmetrical double-stepped beam under axial loading is developed. The beam is composed of two thick segments and three thin segments where a thick segment is between two thin segments, and the longitudinal axis of the thick segment is not colinear with that of the thin segment. The kinetostatic model based on the beam constraint model (BCM) is capable of predicting accurate bending and near-buckling behaviors of the beam. A virtual rigid link neighboring the noncolinear segments is introduced in the BCM to broaden the applicability of the BCM. An analytical formula to represent the critical load of a symmetrical double-stepped beam under axial loading is derived and the value calculated by the formula agrees with the limit load of the asymmetrical double-stepped beam within the elastic range. The investigated beam has potential applications in displacement amplifiers and robotic grippers.

Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

The virtual braid group VBn, the virtual twin group VTn and the virtual triplet group VLn are extensions of the symmetric group Sn, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group Sn are the pure virtual braid group VPn, the pure virtual twin group PVTn and the pure virtual triplet group PVLn, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group PVLn, the commutator subgroup VTn′ of VTn and the commutator subgroup VLn′ of VLn. Our results complete the understanding of these groups, except that of VBn′, for which the existence of a finite presentation is not known for n≥4. We also prove that VLn/PVLn′ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.

Propagation Mechanism and Suppression Strategy of DC Faults in AC/DC Hybrid Microgrid

Due to their efficient renewable energy consumption performance, AC/DC hybrid microgrids have become an important development form for future power grids. However, the fault response will be more complex due to the interconnected structure of AC/DC hybrid microgrids, which may have a serious influence on the safe operation of the system. Based on an AC/DC hybrid microgrid with an integrated bidirectional power converter, research on the interaction impact of faults was carried out with the purpose of enhancing the safe operation capability of the microgrid. The typical fault types of the DC sub-grid were selected to analyze the transient processes of fault circuits. Then, AC current expressions under the consideration of system interconnection structure were derived and, on this basis, we obtained the response results of non-fault subnets under the fault process, in order to reveal the mechanism of DC fault propagation. Subsequently, a current limitation control strategy based on virtual impedance control is proposed to address the rapid increase in the DC fault current. On the basis of constant DC voltage control in AC/DC hybrid microgrids, a virtual impedance control link was added. The proposed control strategy only needs to activate the control based on the change rate of the DC current, without additional fault detection systems. During normal operations, virtual impedance has a relatively small impact on the steady-state characteristics of the system. In the case of a fault, the virtual impedance resistance value is automatically adjusted to limit the change rate and amplitude of the fault current. Finally, a DC fault model of the AC/DC hybrid microgrid was built on the RTDS platform. The simulation and experimental results show that the control strategy proposed in this paper can reduce the instantaneous change rate of the fault state current from 19.1 kA/s to 2.73 kA/s, and the error between the calculated results of equivalent modeling and simulation results was within 5%. The obtained results verify the accuracy of the mathematical equivalent model and the effectiveness of the proposed current limitation control strategy.

On Vassiliev invariants of virtual knots

We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in [1]. In particular, following the theory of quantum invariants we work with ‘rotational’ virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.