Stick Number Research Articles

Lattice stick number 15 is unattainable for non-splittable links

In this paper, we explore mathematical links, defined as closed curves embedded in 3D space. Knot theory studies these structures, which also occur in real-world biopolymers like DNA. Lattice links are links in the cubic lattice. For scientific simulations or statistical studies, links are simplified to lattice links. The lattice stick number, denoted as s L (K), is the minimum number of lattice sticks needed to represent a link K in the cubic lattice. In previous study, it was shown that only two non-trivial knots and six non-splittable links have s L ≤ 14: specifically, sL(212)=8 , sL(31)=sL(212♯212) = sL(623)=sL(633)=12 , sL(412)=13 , and sL(41)=sL(512)=14 . Recent study has further revealed that no knot can have s L = 15. In this paper, we prove that lattice stick number 15 is not attainable for non-splittable links. As a corollary, eleven non-splittable links with s L =16 are presented.

Numerical simulations of fault stick–slip characteristics in different temperature fields at laboratory scale

With the increasing demand for deep engineering development, there is an urgent need to study the evolution and formation mechanisms of the stick–slip process in faults at high temperatures. Although research on fault stick–slip behaviors at room temperature and laboratory scales has yielded some findings, exploring them at high temperatures is challenging, especially when acoustic emission (AE) monitoring is involved. To address this, the research conducted numerical simulations of fault stick–slip characteristics under thermo-mechanical coupling using the discrete element method to solve the problem. This approach aimed to provide insights into fault stick–slip behaviors and AE characteristics across different temperature fields. Subsequently, the evolution of the stress–strain characteristics (the number of stick–slip cycles, slip-initiation stress, slip-initiation stress drop, and maximum stress drop), energy dissipation, and AE characteristics (energy, magnitude, failure mechanism, and b-value) in the stick–slip process were systematically analyzed. The findings of this study aim to address existing shortcomings in AE tests and simulation methods related to fault stick–slip behaviors at high temperatures.

Topology-based optimization of handcuff graphs on 3D lattice

In the field of macromolecular chemistry, handcuff-shaped catenanes and pretzelanes have a conformation consisting of two distinct loops and an edge connecting them. In spatial graph theory, this shape is referred to as a handcuff graph. One topological aspect of interest in these molecular structures involves determining the minimal number of monomers required to create them. In this paper, we focus on a handcuff graph situated in the cubic lattice, which we refer to as a lattice handcuff graph. We explicitly verify that constructing a lattice handcuff graph requires at least 14 lattice sticks, except for the two handcuff graphs: the trivial handcuff graph and the Hopf-linked handcuff graph. Mainly we employ the properly leveled lattice conformation argument, which was developed by the authors to find the lattice stick number of knot-shaped and link-shaped molecules.

Bounds in simple hexagonal lattice and classification of 11-stick knots

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial [Formula: see text]-stick knots in the sh-lattice are the trefoil knot ([Formula: see text]) and the figure-eight knot ([Formula: see text]).

The stick number of rail arcs

Consider two parallel lines [Formula: see text] and [Formula: see text] in [Formula: see text]. A rail arc is an embedding of an arc in [Formula: see text] such that one endpoint is on [Formula: see text], the other is on [Formula: see text], and its interior is disjoint from [Formula: see text]. Rail arcs are considered up to rail isotopies, ambient isotopies of [Formula: see text] with each self-homeomorphism mapping [Formula: see text] and [Formula: see text] onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.

Lattice conformation of theta-curves accompanied with Brunnian property

A theta-curve is an embedding of the Greek letter Θ shaped graph in three-dimensional space. This is a useful physical model for polymer chains since theta-curve motifs are often present in many circular proteins with internal bridges. A Brunnian theta-curve is a nontrivial theta-curve with the property that if we remove any one among three edges, then the remaining knot can be laid in the plane without crossings. We focus on the rigidity of polymer chains with the Brunnian theta-curve shape by using the lattice stick number which is the minimal number of sticks glued end-to-end that are necessary to construct the theta-curve in the cubic lattice. The authors have already shown in a previous research that at least 15 lattice sticks are needed to construct Brunnian theta-curves. In this paper, we improve the lower bound of the lattice stick number for Brunnian theta-curves to 16.

New superbridge index calculations from non-minimal realizations

Previous work [C. Shonkwiler, New computations of the superbridge index, J. Knot Theory Ramifications 29(14) (2020) 2050096] used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. This work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known.

The Effects of Surface Roughness of the Stainless-Steel Anode on Electricity Enhancement of Microbial Fuel Cell

Microorganisms actively catabolize substrate, and bioelectricity is generated. Microbial fuel cells (MFCs) could be utilized as a power generator in small devices. The discovery of species of microorganisms is called Rhodopseudomonas palustris KU-EGAT 13. The experiments have been conducted with the production of electricity from this type of microorganism in a single chamber microbial fuel cell. The study used 4 surface roughness of anode electrode single chamber micro fuel to study the performance of microbial fuel cell effect from the anode. Three stainless steel plates were polished to uniform roughness to the magnitude between 0.05 and 1 µm. After 24 h of experimentation, the rough electrode’s open-circuit voltage (OCV) and power densities were much higher than that produced by the smooth one. Moreover, the smooth surface is higher than the charge-transfer resistance of the rough electrode. The rough surface’s better electrochemical performance is due to denser biofilm grown on the surface, which was observed by scanning electron microscopy (SEM) and figuring out the microbial number in an image using an ImageJ program.
 HIGHLIGHTS
 
 The new discovery of species of microorganisms is called Rhodopseudomonas palustris KU-EGAT 13 which can produce electricity in a single chamber microbial fuel cell
 The performance of MFCs has gotten an effect from the different anode surface roughness.The biomass growth is larger than a rougher anode surface, which contributes to anode’s efficiency
 After observed by SEM, found that when it is in the highest anode surface roughness directed towards the number of microbes stick on the surface
 
 GRAPHICAL ABSTRACT

All prime knots through 10 crossings have superbridge index ≤ 5

This paper gives new upper bounds on the stick numbers of the knots [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], as well as on the equilateral stick number of [Formula: see text]. These bounds imply that the knots [Formula: see text], [Formula: see text], and [Formula: see text] have superbridge index [Formula: see text], completing the project of showing that no prime knots through 10 crossings can have superbridge index larger than 5. The current best bounds on stick number and superbridge index for prime knots through 10 crossings are given in Appendix A.

Topological aspects of theta-curves in cubic lattice* *The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government Ministry of Science and ICT (NRF 2020R1G1A1A01101724, 40%). The second and third authors were supported by Institute for Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.

Knots and embedded graphs are useful models for simulating polymer chains. In particular, a theta curve motif is present in a circular protein with internal bridges. A theta-curve is a graph embedded in three-dimensional space which consists of three edges with shared endpoints at two vertices. If we cannot continuously transform a theta-curve into a plane without intersecting its strand during the deformation, then it is said to be nontrivial. A Brunnian theta-curve is a nontrivial theta-curve that becomes a trivial knot if any one edge is removed. In this paper we obtain qualitative results of these theta-curves, using the lattice stick number which is the minimal number of sticks glued end-to-end that are necessary to construct the theta-curve type in the cubic lattice. We present lower bounds of the lattice stick number for nontrivial theta-curves by 14, and Brunnian theta-curves by 15.

The midterm differential effects of heat-not-burn and conventional cigarettes on coronary flow, vascular function and oxidative stress are independent of nicotine levels

Abstract Introduction Heat-not-burn cigarette (HNBC) constitutes a non-combustible smoke product linked to reduced exposure to carbon monoxide (CO). Methods We compared the effects of HNBC to those of tobacco cigarette (TCig) on endothelial and coronary function as well as on nicotine levels, oxidative stress and platelet activation after 1 month of switching to HNBC. We examined 75 smokers. Of those, 50 were switched to HNBC and 25 continued Tcig for 1 month. At baseline and at 1 month, we assessed a) coronary flow reserve (CFR) by echocardiography b) flow-mediated dilation (FMD) c) Cotinine blood levels, a stable metabolite of nicotine d) malondialdehyde (MDA), a marker of oxidative stress and thromboxane B2 (TXB2), a marker of platelet activation e) the exhaled CO and the number of cigarettes and/or heat stick of HNBC used. Fagerstrom score, a marker of dependence from smoking habit was also calculated. Results Compared to Tcig smoking, switching to HNBC for 1-month improved CO (difference in CO between groups: 10.42 ppm; 95% CI 3.07 to 17.76, p=0.007), FMD (difference in FMD=4.3%; 95% CI: 1.23 to 7.51, p=0.009; and CFR (difference in CFR =0.98; 95% CI: 0.23 to 1.80, p=0.02). MDA and TXB2 concentration significantly decreased in subjects switching to HNBC compared to tobacco smokers (difference MDA=0.38 nmol/L; 95% CI 0.10 to 0.66, p=0.009, 45pg/mL; 95% CI 5.28 to 86.31, p=0.03). None of the aforementioned parameters changed in the control group at 1 month compared to baseline (p>0.05). Cotinine blood levels were similar between the TC cig and HNBC group both at baseline and after one month of use (p>0.05). However, cotinine blood levels at baseline and 1 month correlated with the number cigarettes used at baseline (r=0.45 p=0.04) or the number of heat sticks used at one month (r=0.50, p=0.03) respectively, Baseline Fagerstrom score was also associated with number of heat sticks used at 1 month (r=0.48, p=0.04) and cotinine levels at baseline and at 1 month (p=0.50, p=0.03 and r=0.57, p=0.01 respectively). Conclusions HNBCs exert a less detrimental effect on vascular function platelet activation and oxidative stress than tobacco smoking that is independent of nicotine levels and is linked with the parallel reduction the exhaled of CO. Funding Acknowledgement Type of funding sources: None.

New Stick Number Bounds from Random Sampling of Confined Polygons

The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings.

Stick numbers of Montesinos knots

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

New computations of the superbridge index

The knots [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] all have superbridge index equal to 4. This follows from new upper bounds on superbridge index not coming from the stick number and increases the number of knots from the Rolfsen table for which superbridge index is known from 29 to 49. Appendix A gives the current state of knowledge of superbridge index for prime knots through 10 crossings.

Knots with exactly 10 sticks

We prove that the knots [Formula: see text] and [Formula: see text] both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known.

Lattice stick number of spatial graphs

The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number [Formula: see text] of spatial graphs [Formula: see text] with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number [Formula: see text] [Formula: see text] where [Formula: see text] has [Formula: see text] edges, [Formula: see text] vertices, [Formula: see text] cut-components, [Formula: see text] bouquet cut-components, and [Formula: see text] knot components.

Circuit presentation and lattice stick number with exactly four z-sticks

The lattice stick number [Formula: see text] of a link [Formula: see text] is defined to be the minimal number of straight line segments required to construct a stick presentation of [Formula: see text] in the cubic lattice. Hong, No and Oh [Upper bound on lattice stick number of knots, Math. Proc. Cambridge Philos. Soc. 155 (2013) 173–179] found a general upper bound [Formula: see text]. A rational link can be represented by a lattice presentation with exactly 4 [Formula: see text]-sticks. An [Formula: see text]-circuit is the disjoint union of [Formula: see text] arcs in the lattice plane [Formula: see text]. An [Formula: see text]-circuit presentation is an embedding obtained from the [Formula: see text]-circuit by connecting each [Formula: see text] pair of vertices with one line segment above the circuit. By using a two-circuit presentation, we can easily find the lattice presentation with exactly four [Formula: see text]-sticks. In this paper, we show that an upper bound for the lattice stick number of rational [Formula: see text]-links realized with exactly four [Formula: see text]-sticks is [Formula: see text]. Furthermore, it is [Formula: see text] if [Formula: see text] is a two-component link.

Silk structure rather than tensile mechanics explains web performance in the moth-specialized spider, Cyrtarachne.

Orb webs intercept and retain prey so spiders may subdue them. Orb webs are composed of sticky, compliant spirals of capture silk spun across strong, stiff major ampullate silk threads. Interplay between differences in the mechanical properties of these silks is crucial for prey capture. Most orb webs depend upon insects contacting several radial and capture threads for successful retention. Moths, however, escape quickly from most orb webs due to the sacrificial scales covering their bodies. Cyrtarachne orb webs are unusual as they contain a reduced number of capture threads and moths stick unusually well to single threads. We aimed to determine how the tensile properties of the capture spiral and radial threads spun by Cyrtarachne operate in retention of moth prey. A NanoBionix UTM was used to quantify the material properties of flagelliform and major ampullate threads to test if Cyrtarachne's reduced web architecture is accompanied by improvements in tensile performance of its silk. Silk threads showed tensile properties typical of less-specialized orb-weavers, with the exception of high extensibility in radial threads. Radial thread diameters were 62.5% smaller than flagelliform threads, where commonly the two are roughly similar. We utilized our tensile data to create a finite element model of Cyrtarachne's web to investigate energy dissipation during prey impact. Large cross-sectional area of the flagelliform threads played a key role in enabling single capture threads to withstand prey impact. Rather than extraordinary silk, Cyrtarachne utilizes structural changes in the size and attachment of silk threads to facilitate web function.

Stick number of spatial graphs

For a nontrivial knot [Formula: see text], Negami found an upper bound on the stick number [Formula: see text] in terms of its crossing number [Formula: see text] which is [Formula: see text]. Later, Huh and Oh utilized the arc index [Formula: see text] to present a more precise upper bound [Formula: see text]. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number [Formula: see text] as follows; [Formula: see text]. As a sequel to this research program, we similarly define the stick number [Formula: see text] and the equilateral stick number [Formula: see text] of a spatial graph [Formula: see text], and present their upper bounds as follows; [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are the number of edges and vertices of [Formula: see text], respectively, [Formula: see text] is the number of bouquet cut-components, and [Formula: see text] is the number of non-splittable components.

Lattice stick number of knots**Project supported by the NSFC grants 11531006, 11371367 and 11271290, and the Fundamental Research Funds for the Central Universities 20720160038 and Fujian Province young and middle-aged teacher education research project JA15016.

The minimal number of straight line segments required to construct a polygonal presentation of the knot K in the cubic lattice is called the lattice stick number of the knot K, denoted by . It is known that if the crossing number of K, , satisfies , and the main result of this paper is to improve this to if . Furthermore, we will show that for and which implies that this lower bound cannot be improved except for knots with higher crossing numbers.