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.

Error Uncertainty Analysis in Planar Closed-Loop Structure with Joint Clearances

For planar closed-loop structures with clearances, the angular and positional error uncertainties are studied. By using the vector translation method and geometric method, the boundaries of the errors are analyzed. The joint clearance is considered as being distributed uniformly in a circle area. A virtual link projection method is proposed to deal with the clearance affected length error probability density function (PDF) for open-loop links. The error relationship between open loop and closed loop is established. The open-loop length PDF and the closed-loop angular error PDF both approach being Gaussian distribution if there are many clearances. The angular propagation error of multi-loop structures is also investigated by using convolution. The positional errors of single and multiple loops are both discussed as joint distribution functions. Monte Carlo simulations are conducted to verify the proposed methods.

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.

Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link

Instant center is an important kinematic characteristic which can be used for velocity and singularity analysis, configuration synthesis and dynamics modeling of multi-degree of freedom (multi-DOF) planar linkage. The Aronhold–Kennedy theorem is famous for locating instant centers of four-bar planar linkage, but for single-loop multi-DOF linkages, it fails. Increasing with the number of the links of single-loop multi-DOF planar linkages, the lack of link relationship makes the identification of instant center become a recognized difficulty. This paper proposes a virtual link method to identify instant centers of single-loop multi-DOF planar linkage. First, three types of instant centers are redefined and the instant center identification process graph is introduced. Then, based on coupled loop chain characteristic and definition of instant center, two criteria are presented to convert single-loop multi-DOF planar linkage into a two-loop virtual linkage by adding the virtual links. Subsequently, the unchanged instant centers are identified in the virtual linkage and used to acquire all the instant centers of original single-loop multi-DOF planar linkage. As a result, the instant centers of single-loop five-bar, six-bar planar linkage with several prismatic joints are systematically researched for the first time. Finally, the validity of the proposed method is demonstrated using loop equations. It is a graphical and straightforward method and the application is wide up to single-loop multi-DOF N-bar (N ≥ 5) planar linkage.

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.

Virtual Link Aggregation Network Performance Using MikroTik Bonding

Quality of Service in a network is a big thing that must be resolved and dealt with as best as possible. The limitation of the maximum transfer rate in network devices creates an obstacle in the process of transferring data packets. To maximize the transfer rate in network devices, you can use Virtual Link Aggregation which can offer bandwidth optimization and failover in the network. Link aggregation is a solution in combining several physical links into one logical link. The method used in this research is to consider the allocation of bandwidth, load balancing and failover in the link aggregation. From the results of the link aggregation test using two (2) interface bonding, the results of the bandwidth averages when there is a UPD data packet transfer to 0 bps / 184.9 Mbps, which was previously around 0 bps / 91.6 Mbps. While the result of the bandwidth averages when the TCP data packet transfer occurs is 0 bps / 105.5 Mbps, which was previously around 0 bps / 93.8 Mbps. Link Aggregation using a Mikrotik Router is a solution to produce a larger Throughput Bandwidth by combining two (2) Ethernet Physical Links into one logical link.