Link Diagram Research Articles

Quandle coloring quivers

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.

Exploring the limits of complexity: A survey of empirical studies on graph visualisation

For decades, researchers in information visualisation and graph drawing have focused on developing techniques for the layout and display of very large and complex networks. Experiments involving human participants have also explored the readability of different styles of layout and representations for such networks. In both bodies of literature, networks are frequently referred to as being ‘large’ or ‘complex’, yet these terms are relative. From a human-centred, experiment point-of-view, what constitutes ‘large’ (for example) depends on several factors, such as data complexity, visual complexity, and the technology used. In this paper, we survey the literature on human-centred experiments to understand how, in practice, different features and characteristics of node–link diagrams affect visual complexity.

Classification of Links of Small Complexity in the Thickened Torus

We present a complete table of links in the thickened torus T2 × I of complexity at most 4. The table is constructed by the following method. First, we enumerate all abstract quadrivalent graphs with at most four vertices. Then we consider all inequivalent embeddings of these graphs in the torus. After that each vertex of each of the obtained graphs is replaced by a crossing of one of two possible types specifying the over-strand and the under-strand. The words “over” and “under” are understood in the sense of the coordinate of the corresponding point in the interval I. As a result, we obtain a family of link diagrams in the torus. We propose a number of artificial tricks that essentially reduce the enumeration and help to prove rigorously the completeness of the table. We use a generalized version of the Kauffman polynomial to prove that the obtained links are pairwise inequivalent.

The Jones Polynomial and Functions of Positive Type on the Oriented Jones–Thompson Groups $$\vec {F}$$ F → and $$\vec {T}$$ T →

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones-Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups $\vec{F}$ and $\vec{T}$. Within this framework, we discuss an alternative approach that relies on some partition function interpretation of the Jones polynomial, and also exhibit more examples associated with other link invariants, notably the two-variable Kauffman polynomial and the HOMFLY polynomial. In the unoriented case, extending our previous results, we also show by similar methods that certain evaluations of the Tutte polynomial and of the Kauffman bracket, suitably renormalized, yield functions of positive type on $T$.

Milnor invariants of string links, trivalent trees, and configuration space integrals

We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent “homotopy link diagrams” which corresponds to all finite type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas for products of Milnor invariants from trivalent forests. We also obtain cohomology classes in spaces of link maps from the same data.

A Symmetry Motivated Link Table

Proper identification of oriented knots and 2-component links requires a precise link nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible symmetry types for each link type. The study revisits the methods previously used to disambiguate chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two prime 2-component links with up to nine crossings. Guided by geometrical data, linking number, and the symmetry groups of 2-component links, canonical link diagrams for all but five link types (9 5 2, 9 34 2, 9 35 2, 9 39 2, and 9 41 2) are proposed. We include complete tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also prove a result on the behavior of the writhe under local lattice moves.

Fusion and monodromy in the Temperley-Lieb category

Graham and Lehrer (1998) introduced a Temperley-Lieb category \ctl whose objects are the non-negative integers and the morphisms in \Hom(n,m) are the link diagrams from nn to mm nodes. The Temperley-Lieb algebra \tl n is identified with \Hom(n,n). The category \ctl is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on \ctl. We introduce a module category \modtl whose objects are functors from \ctl to \mathsf{Vect}_{\mathbb C}𝖵𝖾𝖼𝗍ℂ and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for \ctl to induce the structure of a ribbon category on \modtl(\beta=-q-q^{-1}), when qq is not a root of unity. We discuss how the braiding on \ctl and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.

Using an innovative tool in science education: examining pre-service elementary teachers’ evaluation levels on the topic of wetlands

ABSTRACTToday’s world requires citizens to make informed decisions by critically evaluating evidence and alternative explanations. The purpose of this study was to explore pre-service elementary teachers’ capability of building a model–evidence link, their evaluation levels on the topic of wetlands, and their evaluations of the trustworthiness of the sources of evidence. In order to achieve this goal, we used the Turkish-adapted version of Model-Evidence Link (MEL) diagrams, an instructional tool that provides insights into the ability of learners to build model–evidence links and their explanations for these links. The analysis of participant responses revealed an accurate understanding of model–evidence links; however, participants showed a misinterpretation when evidence has nothing to do with the model. The findings indicated that the participants usually explain these links in descriptive and/or relational terms. The analysis also showed that the participants listed various criteria for evaluating the trustworthiness of the sources of evidence; however, the percentage of each was low. The implications of this study suggest a need for using MEL-like scaffolds in various long-term educational settings as well as a need for purposeful and systematic efforts to enable learners to evaluate sources of evidence for different subjects.

Coherent double coverings of virtual link diagrams

A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links. In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram. We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent. We relate this method to some invariants of virtual links.

The self-smoothing number of knots and links

We call smoothing a self-crossing point of an oriented link diagram self-smoothing. By self-smoothing repeatedly, we obtain an oriented link diagram without self-crossing points. In this paper, we show that every knot has an oriented diagram which becomes a two-component oriented link diagram without self-crossing points by a single self-smoothing.

The link transmission model with variable fundamental diagrams and initial conditions

ABSTRACTThe link transmission model is a macroscopic network traffic flow simulation tool based on Lighthill–Whitham–Richards theory. While its efficiency and accuracy are superior to the well-known cell transmission model, applications of its current numerical formulations are limited by the inability to apply changes to the fundamental diagrams of links within a simulation and the need to start the simulation with an empty network. We resolve both limitations by developing a methodology for initialising the discrete-time link model with a non-empty initial condition and for computing within-link densities during the simulation, which can then serve as an initial condition for continued simulation with a new fundamental diagram. Since the computation of within-link densities is algebraic, no new numerical errors are introduced. Optional support for multiple commodities, subcritical delays and platoon dispersion, are retained. The resulting model is demonstrated on a motorway corridor network with variable speed limits and dynamic lane management.

Tangle blocks in the theory of link invariants

The central discovery of 2d conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire huge Hilbert space of states. Somewhat similar, when a link diagram is glued from tangles, the link polynomial is a multilinear combination of tangle blocks summed over just a few representations of intermediate states. This turns to be a powerful approach because the same tangles appear as constituents of very different knots so that they can be extracted from simpler cases and used in more complicated ones. So far this method has been technically developed only in the case of arborescent knots, but, in fact, it is much more general. We begin a systematic study of tangle blocks by detailed consideration of some archetypical examples, which actually lead to non-trivial results, far beyond the reach of other techniques. At the next level, the tangle calculus is about gluing of tangles, and functorial mappings from Hom(tangles). Its main advantage is an explicit realization of multiplicative composition structure, which is partly obscured in traditional knot theory.

Graphical virtual links and a polynomial for signed cyclic graphs

For a signed cyclic graph [Formula: see text], we can construct a unique virtual link [Formula: see text] by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link [Formula: see text] is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial [Formula: see text] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between [Formula: see text] of a signed cyclic graph [Formula: see text] and the bracket polynomial of one of the virtual link diagrams associated with [Formula: see text]. Finally, we give a spanning subgraph expansion for [Formula: see text].

Robust Diagnosis of Industrial Systems by Bond graph Model

This paper aims to solve a research problem by a robust diagnosis of a hydraulic system with sand filter by approach of graph of connection using fractional linear transformations (BG-LFT). The method we develop is based on the use of analytic redundant relations by a bond graph graph model (BG-ARRs). These relationships not only allow the detection and isolation of defects on the various elements of the system, but also the location by structural and causal analysis. The results suggest that the use of the link diagram model for a valve fault (eg an R1 valve is blocked), figures 9.a), 9.b) and 9.c) show that the residual models r1 and r2 become non-zero, so the flow and pressure levels are zero at reservoir C2. These values mean that these residues are sensitive to the variation of the flow rate at the level of the valve R1, which is confirmed by the theoretical results presented in table 2. The simulation of the system is carried out by the software dedicated to the bond graph approach

Structure and Enumeration of K4-minor-free links and link diagrams

We study the class L of link types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimal number of crossings or edges in a projection of L∈L. Further, we enumerate (both exactly and asymptotically) all connected K4-minor-free link diagrams, all minimal connected K4-minor-free link diagrams, and all K4-minor-free diagrams of the unknot.

Simplicial homotopy theory, link homology and Khovanov homology

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

Splitting number of augmented links

We completely determine the splitting number of augmented links arising from knot and link diagrams in which each twist region has an even number of crossings. In the case of augmented links obtained from knot diagrams, we show that the splitting number is given by the size of a maximal collection of Boromean sublinks, any two of which have one component in common. The general case is stablished by considering the linking numbers between components of the augmented links. We also discuss the case when the augmented link arises from a link diagram in which twist regions may have an odd number of crossings.

Link invariants derived from multiplexing of crossings

We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers $m_{i}$ $(i=1,\ldots,n)$ and an ordered $n$-component virtual link diagram $D$, a new virtual link diagram $D(m_{1},\ldots,m_{n})$ is obtained from $D$ by the multiplexing of all crossings. For welded isotopic virtual link diagrams $D$ and $D'$, $D(m_{1},\ldots,m_{n})$ and $D'(m_{1},\ldots,m_{n})$ are welded isotopic. From the point of view of classical link theory, it seems very interesting that $D(m_{1},\ldots,m_{n})$ could not be welded isotopic to a classical link diagram even if $D$ is a classical one, and new classical link invariants are expected from known welded link invariants via the multiplexing of crossings.

NMR-based metabolic toxicity of low-level Hg exposure to earthworms

Mercury is a globally distributed toxicant to aquatic animals and mammals. However, the potential risks of environmental relevant mercury in terrestrial systems remain largely unclear. The metabolic profiles of the earthworm Eisenia fetida after exposure to soil contaminated with mercury at 0.77 ± 0.09 mg/kg for 2 weeks were investigated using a two-dimensional nuclear magnetic resonance-based (1H-13C NMR) metabolomics approach. The results revealed that traditional endpoints (e.g., mortality and weight loss) did not differ significantly after exposure. Although histological examination showed sub-lethal toxicity in the intestine as a result of soil ingestion, the underlying mechanisms were unclear. Metabolite profiles revealed significant decreases in glutamine and 2-hexyl-5-ethyl-3-furansulfonate in the exposed group and remarkable increases in glycine, alanine, glutamate, scyllo-inositol, t-methylhistidine and myo-inositol. More importantly, metabolic network analysis revealed that low mercury in the soil disrupted osmoregulation, amino acid and energy metabolisms in earthworms. A metabolic net link and schematic diagram of mercury-induced responses were proposed to predict earthworm responses after exposure to mercury at environmental relevant concentrations. These results improved the current understanding of the potential toxicity of low mercury in terrestrial systems.

Introduction to disoriented knot theory

Abstract This paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.