The knot invariant Υ using grid homologies

According to the idea of Ozsváth, Stipsicz and Szabó, we define the knot invariant [Formula: see text] without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that [Formula: see text], as introduced this way, is a well-defined knot invariant. We reprove some important propositions using the new techniques, and show that [Formula: see text] provides a lower bound on the unknotting number.

Minimal grid diagrams of 11 crossing prime alternating knots

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentations with 13 arcs.

Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process such as the topoisomerase-mediated preferential unknotting of DNA molecules. We model here topoisomerase-mediated passages of double-stranded DNA segments through each other using the formalism of grid diagrams. We show that this grid diagram–based modeling approach captures the essence of the preferential unknotting mechanism, based on topoisomerase selectivity of hooked DNA juxtapositions as the sites of intersegmental passages. We show that the grid diagram–based approach provides an important, new, and computationally convenient framework for investigating entanglement in biopolymers.

The influence of SRA programming on algorithmic thinking and self-efficacy using Lego robotics in two types of instruction

This study investigates the development of algorithmic thinking as a part of computational thinking skills and self-efficacy of primary school pupils using programmable robots in different instruction variants. Computational thinking is defined in the context of twenty-first century skills and describes processes involved in (re)formulating a problem in a way that a computer can process it. Programming robots offers specific affordances as it can be used to develop programs following a Sense-Reason-Act (SRA) cycle. The literature provides evidence that programming robots has the potential to enhance algorithmic thinking as a component of computational thinking. Specifically there are indications that pupils who use SRA-programming learn algorithmic skills better and achieve a higher level of self-efficacy in an open, scaffold learning environment than through direct instruction. In order to determine the influence of the instruction variant used, an experimental research design was made in which pupils solved algorithm-based mathematical problems (grid diagrams) in a preliminary measurement and their self-efficacy determined via a questionnaire. As an intervention, pupils learn to solve programming issues in pairs using “Lego NXT” robots and “Mindstorms” software in two instruction variants. The post-measurement consists of a Lego challenge, solving mathematical problems (grid diagrams), and a repeated self-efficacy questionnaire. This research shows an increase of our measures on algorithmic thinking dependent on the amount of SRA usage (though not significant). Programming using the SRA-cycle can be considered as the cause of the measured effect. The instruction variant used during the robotic intervention seems to play only a marginal role.

Continuous Fuzzy Kano Model and Fuzzy AHP Model for Aesthetic Product Design: Case Study of an Electric Scooter

The fuzzy Kano model (FKM) adopts a linear scoring system that cannot accurately reflect the relationship between users’ needs and satisfaction and may underestimate the importance of users’ evaluation and needs. In this paper, we propose a preference-based evaluation-fuzzy-quantification method to determine the priority of the development of attractive factors of an electric scooter. In the evaluation analysis stage, the evaluation grid diagrams of all the interviewees are systemized. In the fuzzy computing stage, the continuous fuzzy Kano model (C-FKM) combined with the fuzzy analytic hierarchy process (FAHP) is developed to determine the priority of the development of attractive factors. By processing the ambiguity of users’ needs, the C-FKM can obtain a more accurate representation of users’ needs than the FKM. The opinions of 20 experts are integrated using the similarity aggregation method (SAM); then the FAHP is applied to calculate the weight of each evaluation criterion. Lastly, we use quantitative analysis to discover the important and specific characteristics that influence the attractiveness of an electric scooter. Research shows that Kansei images with attractive qualities are reliable and may be the first choice for satisfying the perceptual demands of consumers and can provide reference for the related research.

Grid diagram for singular links

In this paper, we define the set of singular grid diagrams [Formula: see text] which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on [Formula: see text] which induce the bijection onto each singular object. This is an extension of the known result of Ng–Thurston [Grid diagrams, braids, and contact geometry, in Proc. Gökova Geometry-Topology Conf. 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 120–136] for nonsingular links and braids.

On the arc index of cable links and Whitehead doubles

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of [Formula: see text]-cable links and Whitehead doubles of all prime knots with up to eight crossings.