Link Diagram Research Articles

Tribracket Modules

Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modules and use these structures to enhance the tribracket counting invariant. We provide examples to illustrate the computation of the invariant and show that the enhancement is proper.

Ineffective sets and the region crossing change operation

Region crossing change (RCC) is an operation on link diagrams in which all crossings incident to a selected region are changed. Two diagrams are called RCC-equivalent if one can be transformed to the other via a sequence of RCCs. RCC is an unknotting operation but not an unlinking operation. A set of regions of a diagram is called ineffective if RCCs at every region in that set have no net effect on the crossings of the diagram. The main result of this paper is a construction of the complete collection of ineffective sets for any link diagram. This involves a combination of linear algebraic and diagrammatic techniques including a generalization of checkerboard shading called tricoloring. Using this construction of ineffective sets, we provide sharp upper bounds on the maximum number of RCCs needed to transform between RCC-equivalent knot diagrams and reduced 2- and 3-component link diagrams with fixed underlying projections.

Classification of prime links in the thickened torus having crossing number 5

The goal of this paper is to tabulate all prime links in the thickened torus [Formula: see text] having diagrams having crossing number 5. First, we construct a table of prime projections of links on the torus [Formula: see text] having exactly 5 crossings. To this end, we enumerate abstract quadrivalent graphs of special type and consider all possible embeddings of the graphs into the torus [Formula: see text] in order to construct prime projections. Then, we prove that all obtained projections are inequivalent. Second, we use the list of prime projections to construct a table of diagrams of prime links in the torus [Formula: see text]. In order to prove that all those links are inequivalent, we use two modifications of the Kauffman bracket. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed tables.

Virtual links which are equivalent as twisted links

A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly nonorientable) closed surfaces. By definition, equivalent virtual links are also equivalent as twisted links. In this paper, we discuss when two virtual links are equivalent as twisted links, and give a necessary and sufficient condition for this to be the case.

The Cone Structure Theorem

Abstract We consider the topological classification of finitely determined map germs $f:(\mathbb {R}^n,0)\to (\mathbb {R}^p,0)$ with $f^{-1}(0)\neq \{0\}$. Associated with $f$ we have a link diagram, which is well defined up to topological equivalence. We prove that $f$ is topologically $\mathcal {A}$-equivalent to the generalized cone of its link diagram.

Wirtinger numbers for virtual links

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

The calculation of topological structures of strands-based DNA trigonal bipyramids

With the rapid development of DNA nanotechnology, strands-based DNA polyhedra have been reported and widely applied in chemical biology, drug delivery and materials science. A fundamental problem is to determine which topological structures such a DNA polyhedron will allow. Our goal in this paper is to determine all permissible nontrivial topological structures for DNA trigonal bipyramids with double-helical edges, which have been partially realized by one-step self-assembly of multiple DNA single-strands designed rationally. Here oriented trigonal bipyramid links (OTB links) are constructed as the mathematical models for DNA trigonal bipyramids and are further classified into 451 link types. In this process, an OTBL program is designed in the Python programming language to generate all OTB link diagrams and also to remove the same topological structures produced by the rotational symmetry of trigonal bipyramid. Our result gives a list of candidates for further synthesized DNA trigonal bipyramid prisms with required topological structures.

Teaching socio-scientific issues through evidence-based thinking practices: Appropriateness, benefits, and challenges of using an instructional scaffold

Teachers are expected to improve their students’ analytical thinking and decision-making skills through evidence-based thinking and critical evaluation processes. In this study, a three-hour workshop was conducted to investigate science teachers’ views about teaching socio-scientific issues through argumentation and introducing an instructional scaffold, Model-Evidence Link diagrams to promote the use of argumentation and critical evaluation in science classrooms. 125 science teachers, who were working in public schools of an urban area in Turkey participated in the workshop. Findings revealed that 90% of the participants stated that the use of MEL diagram is appropriate for science teaching. Promoting higher order thinking skills was the highest benefit, whereas the need for time for the development and implementation of the material was the greatest challenge for the use of the MEL diagrams in science classrooms. This study contributes to the literature on teaching socio-scientific issues, especially through argumentation, evidence-based thinking, and critical evaluation.

Cyclic coverings of virtual link diagrams

A virtual link diagram is called mod [Formula: see text] almost classical if it admits an Alexander numbering valued in integers modulo [Formula: see text], and a virtual link is called mod [Formula: see text] almost classical if it has a mod [Formula: see text] almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod [Formula: see text] almost classical virtual link diagram from a given virtual link diagram, which we call an [Formula: see text]-fold cyclic covering diagram. The main result is that [Formula: see text]-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus, we have a well-defined map from the set of virtual links to the set of mod [Formula: see text] almost classical virtual links. Some applications are also given.

Biquandle virtual brackets

We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring [Formula: see text] using virtual crossings as smoothings, these invariants take the form of multisets of elements of [Formula: see text] and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications 26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.

Multi-tribrackets

We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular, we consider the case of component multi-tribrackets which have different tribracket operations at single-component crossings and multi-component crossings. We provide examples to show that the resulting counting invariants can distinguish links which are not distinguished by the counting invariants associated to the standard tribracket coloring. We reinterpret the results of [S. Nelson and S. Pico. Virtual tribrackets, preprint (2018), arXiv:1803.03210] in terms of multi-tribrackets and consider future directions for multi-tribracket theory.

An Argumentative Tool for Facilitating Critical Evaluation

The aim of the present study is to explore pre-service elementary teachers’ evaluations of the evidence and models and their positions on a socio-scientific topic, namely genetically modified organisms (GMOs), after evaluating them through Model Evidence Link (MEL) diagrams. The findings of this study show that the participants mostly constructed accurate evidence-model links. However, they mostly generated inaccurate relationships between the model and the evidence when the evidence had nothing to do with the model. The results also show that the participants mostly evaluated the model-evidence relationships at the descriptive and relational levels. The participants were able to construct accurate links; however, they could not critically evaluate the model-evidence links. Finally, although the participants read supportive, opposing and irrelevant evidence about the benefits of GMOs, and discussed them with their group mates, most of them still thought that GMOs are not beneficial to society. Furthermore, none of them stated the belief that GMOs are beneficial to society.

Investigation of Preservice Teachers’ Model-Evidence Link Evaluation Levels Related to GDO

Although genetically modified organisms (GMOs) are scientific issues, their disadvantages as well as advantages are often asserted in society. In other words, they are controversial issues in society. In this context, individuals need to make informed decisions about GMOs which are part of socio-scientific issues. Making informed decisions by critically evaluating on controversial issues is a significant skill. Therefore this study aims to detect pre-service science teachers’ evaluation level of model-evidence links. The study was a descriptive survey that uses a qualitative data source and research group consisted of junior students of Science Education Department in a public university in Turkey. The Model-Evidence Link (MEL) diagram developed by the high school science teachers who attended a workshop organized Lombardi, Sibley and Carroll (2013) and adapted to GMO by the same teachers was used as the data collection tool. A rubric developed by Lombardi, Bickel, Brandt and Burg (2016) and adapted to this study by the authors of the present paper was used for analyzing the data. The analyses indicated that pre-service teachers evaluated model-evidence links on the topic of GMOs generally in descriptive and relational level, however, rarely in critical level.

Growth rate of quantum knot mosaics

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot (m, n)-mosaic is an $$m \times n$$ array of 11 mosaic tiles representing a knot or a link diagram by adjoining properly. The total number $$D_{m,n}$$ of knot (m, n)-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, the first author showed the existence of the knot mosaic constant $$\delta = \lim _{m, n \rightarrow \infty } (D_{m,n})^{\frac{1}{mn}}$$ and proved $$4 \le \delta \le \frac{5+ \sqrt{13}}{2} = 4.302\cdots $$ by developing an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. In this paper, we give a simpler proof of the lower bound and improve the upper bound of the knot mosaic constant as $$\begin{aligned} 4 \le \delta \le 4.113\cdots \end{aligned}$$by introducing two new concepts: quasimosaics and cling mosaics.

An unknotting index for virtual links

Given a virtual link diagram D, we define its unknotting index U(D) to be minimum among (m,n) tuples, where m stands for the number of crossings virtualized and n stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram we provide a lower bound for unknotting index of a virtual link. Then using warping degree of a diagram, we obtain an upper bound. Both these bounds are applied to find unknotting index for virtual links obtained from pretzel links by virtualizing some crossings.

Virtual tribrackets

We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.

When can a link be obtained from another using crossing exchanges and smoothings?

Let L be a fixed link. Given a link diagram D, is there a sequence of crossing exchanges and smoothings on D that yields a diagram of L? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if L is a prime link with crossing number at most 5, then there is an algorithm that answers this question in polynomial time. We show that the same holds for all torus links T2,m and all twist knots.

Local biquandles and Niebrzydowski's tribracket theory

We introduce a new algebraic structure called local biquandles and show how colorings of oriented classical link diagrams and of broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles and show that it is isomorphic to Niebrzydowski's tribracket (co)homology. This implies that Niebrzydowski's (co)homology theory can be interpreted similarly as biquandle (co)homology theory. Moreover through the isomorphism between two cohomology groups, we show that Niebrzydowski's cocycle invariants and local biquandle cocycle invariants are the same.

Arrow calculus for welded and classical links

We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations, which can be seen as `higher order Gauss diagrams'. This Arrow calculus is used to develop an analogue of Habiro's clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a 'realization' of Polyak's algebra of arrow diagrams at the welded level, and leads to a characterization of finite type invariants of welded knots and long knots. As a corollary, we recover several topological results due to K. Habiro and A. Shima and to T. Watanabe on knotted surfaces in 4-space. We also classify welded string links up to homotopy, thus recovering a result of the first author with B. Audoux, P. Bellingeri and E. Wagner.

The braid index of reduced alternating links

AbstractIt is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and only if GS(D) contains no edges of weight one. Here GS(D), called the Seifert graph of D, is an edge weighted simple graph obtained from D by identifying each Seifert circle of D as a vertex of GS(D) such that two vertices in GS(D) are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in D and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well-known MFW inequality, which states that the a-span of the HOMFLY polynomial of L is a lower bound of 2b(L)−2, as well as the result of Yamada relating the minimum number of Seifert circles over all link diagrams of L to b(L).