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We develop topological methods for characterizing the relationship between polymer chain entanglement and bulk viscoelastic responses. We introduce generalized Linking Number and Writhe characteristics that are applicable to open linear chains. We investigate the rheology of polymeric chains entangled into weaves with varying topologies and levels of chain density. To investigate viscoelastic responses, we perform non-equilibrium molecular simulations over a range of frequencies using sheared Lees⁻Edwards boundary conditions. We show how our topological characteristics can be used to capture key features of the polymer entanglements related to the viscoelastic responses. We find there is a linear relation over a significant range of frequencies between the mean absolute Writhe W r and the Loss Tangent tan ( δ ) . We also find an approximate inverse linear relationship between the mean absolute Periodic Linking Number L K P and the Loss Tangent tan ( δ ) . Our results show some of the ways topological methods can be used to characterize chain entanglements to better understand the origins of mechanical responses in polymeric materials.

We give a criterion for distinguishing a prime knot K in S 3 from every other knot in S 3 using the finite quotients of π 1 (S 3 ∖ K). Using recent work of Baldwin–Sivek, we apply this criterion to the hyperbolic knots 5 2 , 1 5 n 4 3 5 2 2 , and the three-strand pretzel knots P (− 3 , 3 , 2 n + 1) for every integer n. [ABSTRACT FROM AUTHOR]

The N‐terminal half of the giant cytoskeletal protein obscurin is comprised of more than 50 Ig‐like domains, arranged in tandem. Domains 18–51 are connected to each other through short 5‐residue linkers, and this arrangement has been previously shown to form a semi‐flexible rod in solution. Domains 1–18 generally have slightly longer ~7 residue interdomain linkers, and the multidomain structure and motion conferred by this kind of linker is understudied. Here, we use NMR, SAXS, and MD to show that these longer linkers are associated with significantly more domain/domain flexibility, with the resulting multidomain structure being moderately compact. Further examination of the relationship between interdomain flexibility and linker length shows there is a 5 residue "sweet spot" linker length that results in dual‐domain systems being extended, and conversely that both longer or shorter linkers result in a less extended structure. This detailed knowledge of the obscurin N terminus structure and flexibility allowed for mathematical modeling of domains 1–18, which suggests that this region likely forms tangles if left alone in solution. Given how infrequently protein tangles occur in nature, and given the pathological outcomes that occur when tangles do arise, our data suggest that obscurin is likely either significantly scaffolded or else externally extended in the cell. [ABSTRACT FROM AUTHOR]

Models for the elastic material properties of wood aim to predict the mechanical response of wooden structures to external loading. Traditionally, the variability of these properties in trees is described by a taxonomy of growth defects that is typically based on visual indicators in the material. This includes curvature, knots, and spiral grain models. Existing meso- and macro-scale models fail to describe the uncertainty connected to the local heterogeneity of the material. In this paper, we propose a novel meso-scale model that describes the natural variability of Norway spruce morphology and material properties based on random field theory. Our approach removes the need for a taxonomy of growth defects and enables uncertainty quantification of the stiffness and density in a straightforward fashion using simulations. This may enhance confidence for stiffness-graded applications, where the dynamic resonant behavior of wood structures is relevant and growth anomalies are present. Further, our stochastic models can generate images that realistically mimic wood patterns, which is relevant for applications like synthetic wood panels and flooring. [Display omitted] • Validated meso-scale material model for wood using fiber direction and density. • Stochastic model for natural growth features: spiral grain, taper, and curvature. • Probabilistic model for branch knot distribution in the growth direction. • Generation of images that mimic real wood annual ring patterns. [ABSTRACT FROM AUTHOR]

We automate the process of machine learning correlations between knot invariants. For nearly 200 000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural network on the input invariants. Correlation between invariants is measured by the accuracy of the neural network prediction, and bipartite or tripartite correlations are sequentially filtered from the input invariant sets so that experiments with larger input sets are checking for true multipartite correlation. We rediscover several known relationships between polynomial, homological, and hyperbolic knot invariants, while also finding novel correlations which are not explained by known results in knot theory. These unexplained correlations strengthen previous observations concerning links between Khovanov and knot Floer homology. Our results also point to a new connection between quantum algebraic and hyperbolic invariants, similar to the generalized volume conjecture.

The main goal of this article is to determine the nature of a knot set named twotwist (m, n), where m, n are whole positive numbers, using the abelianization of its fundamental group on S3. First, a basic theory to define the concept of knots as topological objects was established. Then, a detailed description of a process named "Mechanical wheel", which was created and developed in this article to identify such abelianization, was provided. Lastly, a description of each twotwist for all m and n was attained. [ABSTRACT FROM AUTHOR]

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for 3 -stick unknots with ribbon linking numbers ± 1 and ± 3 , and we prove that the minimum folded ribbonlength for n -gons with obtuse interior angles is achieved when the n -gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number ± n is bounded from above by 2 n. [ABSTRACT FROM AUTHOR]

We introduce a framework for adsorption of a single polymer in which the topology of the polymer is quenched before adsorption, in contrast to more standard adsorption models having annealed topology. Our “topology” refers either to the precise branching structure of a branched polymer (in any dimension), or else to the knot type of a ring polymer in three dimensions. The quenched topology is chosen uniformly at random from all lattice polymers of a given size in one of four classes (lattice animals, trees, combs, or rings), and we then consider adsorption of the subclass of configurations that have the quenched topology. When the polymer-surface attraction increases without bound, the quenched topological structure keeps a macroscopic fraction of monomers off the surface, in contrast with annealed models that asymptotically have 100% of monomers in the surface. We prove properties of the limiting free energy and the critical point in each model, although important open questions remain. We pay special attention to the class of comb polymers, which admit some rigorous answers to questions that otherwise remain open. Since the class of all combs was not previously examined rigorously in full generality, we also prove the existence of its growth constant and its limiting free energy for annealed adsorption. [ABSTRACT FROM AUTHOR]

Copyright of Wood Industry / Drvna Industrija is the property of Drvna Industrija and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Background: While poplar (Populus spp.) is an important source of raw materials for the wood processing sector in many regions of the world, in New Zealand it has a reputation for producing poor grade recoveries of sawn timber that is unsuitable for structural applications. However, the 'Kawa' poplar clone (Populus deltoides Marshall x P. yunnanensis Dode), which has relatively high wood density, could yield structural timber. This, along with evidence demonstrating building code compliance, would improve utilisation options for this species in New Zealand. Methods: Sawn timber conversion and grade recovery were quantified for a 28-year-old pruned stand of 'Kawa' poplar grown in Northland, New Zealand. A sample of 90 mm x 45 mm structural boards were tested to determine their mechanical properties and the resulting strength class. Boron preservative retention and penetration were measured to determine whether timber could be treated to the level required under New Zealand's building standards. Density, modulus of elasticity and modulus of rupture were assessed on small defect-free specimens taken from different radial and vertical positions within trees to determine intra-stem and inter-stem variation in these properties. Results: The overall conversion of logs to sawn timber was 53%, with approximately 94% of this recovery consisting of graded timber. The most common sources of downgrade were knots, pruning wounds, and end-splits. Approximately 70% of the sawn boards were graded as clears, with smaller recoveries of cladding and structural boards. The average length of clear section was approximately 2.5 m. Mechanical testing of structural boards demonstrated that they have characteristic values sufficient to meet the requirements for the SG10 strength class. Preservative treatment achieved the H1.2 specification. Density, modulus of elasticity and modulus of rupture were all higher in specimens cut from the outside of the log compared with those taken from near the pith at all heights up the stem. Conclusions: Mechanical properties and boron treatment results indicate suitability for structural applications in accordance with New Zealand's building code. 'Kawa' poplar also produced high grade recoveries suggesting potential for commercial sawn timber production, especially for structural appearance products. [ABSTRACT FROM AUTHOR]

The main goal of this article is to determine the nature of a knot set named twotwist (m, n), where m, n are whole positive numbers, using the abelianization of its fundamental group on S3. First, a basic theory to define the concept of knots as topological objects was established. Then, a detailed description of a process named "Mechanical wheel", which was created and developed in this article to identify such abelianization, was provided. Lastly, a description of each twotwist for all m and n was attained. [ABSTRACT FROM AUTHOR]

Changes in the condition of existing timber structures can be caused by fatigue or biological attack, among other things. Replacing damaged timber is still very expensive, so it seems more advisable to repair or reinforce damaged elements. Therefore, in order to improve the static performance analysis of timber structures, reinforcement applications in timber elements are necessary. In this experimental study, technical-scale glulam beams measuring 82 × 162 × 3650 mm, which were reinforced with carbon strands and carbon laminates, were tested in flexure. A four-point bending test was used to determine the effectiveness of the reinforcement used in the timber beams. Internal strengthening (namely, glued carbon cords placed into cut grooves in the last and penultimate lamella) and an external surface of near-surface mounted (NSM) carbon laminates glued to the bottom surface of the beam were used to reinforce the laminated ashlar beams. As a result of this study, it was found that the bending-based mechanical properties of ash wood beams reinforced with carbon fibre-reinforced polymer composites were better than those of the reference beams. In this work, the beams were analysed in terms of the reinforcement variables used and the results were compared with those for the beams tested without reinforcement. This work proves the good behaviour of carbon fibre reinforced plastic (CFRP—Carbon fibre reinforced polymer) cords when applied to timber beams and carbon laminates. This study illustrated the different reinforcement mechanisms and showed their structural properties. Compared to the reference samples, it was found that reinforcement with carbon strings or carbon laminates increased the load-bearing capacity, flexural strength and modulus of elasticity, and reduced the amount of displacement of the timber materials, which is an excellent alternative to the use of ashlar and, above all, inferior grade materials due to the current shortage of choice grade. Experimental results showed that, with the use of carbon fibre (carbon cords SikaWrap® FX-50 C—Sika Poland Sp. z o.o., Warsaw), the load bearing capacity increased by 35.58%, or with carbon cords SikaWrap® FX-50 C and carbon laminates S&P C-Laminate type HM 50/1.4 - S&P Poland Sp. z o.o., Malbork, by 45.42%, compared to the unreinforced beams. [ABSTRACT FROM AUTHOR]

This study aimed to evaluate the mechanical properties determined in a 4-point bending test of beams made of lumber from which knots had been locally removed and the resulting loss replaced with sound wood. Three sets of beams were prepared, which differed in the number of layers/lamellas and the position of the lamellas from which edge knots were removed. All the lamellas used in the tests were subjected to a modulus of elasticity assessment. In addition to the distribution of defects, it determined the position of a given piece in the beam structure. The tests showed that high mechanical properties could characterise the beams produced in this way, i.e., a modulus of elasticity close to 12 kN/mm2 and a strength above 40 N/mm2, if the lamellas without knots were located below the outer tension lamella. Significantly better results were obtained when PUR glue was used in the inserts rather than MUF. In this case, beams with an improved outer lamella in the tension zone using semi-circular inserts glued with PUR glue had an average strength of 34.6 N/mm2. [ABSTRACT FROM AUTHOR]

A subtle interplay between the flexibility of the 2,2'-bipyridyl-based diamine and the denticity of the coordination domain formed upon self-assembly enabled the formation of four distinct topologies stabilised by [Ag···Ag]2+ pairs. The reactions utilising 2,6-diformylpyridine resulted in the formation of silver(I)-stabilised molecular tweezer, trefoil knot, and Solomon link. The 1,8-naphthyridine-based dialdehyde promoted the formation of [2]catenanes and trefoil knot, demonstrating very close AgI···AgI distances. Two of the studied assemblies demonstrated interesting luminescent properties in the solid state., (© 2024 Wiley‐VCH GmbH.)

We investigate knotting probabilities of long double-stranded DNA strands in a coarse-grained Kratky-Porod model for DNA with Monte Carlo simulations. Various ionic conditions are implemented by adjusting the effective diameter of monomers. We find that the occurrence of knots in DNA can be reinforced considerably by high salt conditions and confinement between plates. Likewise, knots can almost be dissolved completely in a low salt scenario. Comparisons with recent experiments confirm that the coarse-grained model is able to capture and quantitatively predict topological features of DNA and can be used for guiding future experiments on DNA knots.

Inspired by Dunitz's treatment of the geometry and flexibility of cyclohexane, we describe some ways in which hexagons and octahedra can become self‐entangled and linked. Connections to topics such as flexibility of frameworks, tensegrity structures, and Borromean rings are indicated. [ABSTRACT FROM AUTHOR]

We performed molecular dynamics simulations of DNA polymer chains confined in helical nano-channels under compression in order to explore the potential of knot-factories with helical geometry to produce knots with a preferred handedness. In our simulations, we explore mutual effect of the confinement strength and compressive forces in a range covering weak, intermediate and strong confinement together with weak and strong compressive forces. The results find that while the common metrics of polymer chain in cylindrical and helical channels are very similar, the DNA in helical channels exhibits greatly different topology in terms of chain knottedness, writhe and handedness of knots. The results show that knots with a preferred chirality in terms of average writhe can be produced by using channels with a chosen handedness. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox. [ABSTRACT FROM AUTHOR]

We consider -difference equations for colored Jones polynomials. These polynomials are invariants for the knots and their asymptotics plays an important role in the famous Volume Conjecture (VC) for the complement of the knot to the sphere. We study WKB asymptotic behavior of the th colored Jones polynomial at the point when and tends to infinity and limit of belongs to . We state a Theorem on asymptotic expansion of general solutions of the -difference equations. For the partial solutions, corresponding to the colored Jones polynomials, using some heuristic and numeric consideration, we suggest a conjecture on their WKB asymptotics. For the special knots under consideration, this conjecture is in accordance with the VC. [ABSTRACT FROM AUTHOR]

The conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of N≫1 edges has only logarithmic, and not power law corrections to the known leading exponential law: Punknot(N)∼exp−N/N0+o(lnN) with N0 being referred to as the random knotting length. This conjecture is consistent with the numerical result of 2010 by Baiesi, Orlandini, and Stella.

The accidental untying of a shoelace while walking often occurs without warning. In this paper, we discuss the series of events that lead to a shoelace knot becoming untied. First, the repeated impact of the shoe on the floor during walking serves to loosen the knot. Then, the whipping motions of the free ends of the laces caused by the leg swing produce slipping of the laces. This leads to eventual runaway untangling of the knot. As demonstrated using slow-motion video footage and a series of experiments, the failure of the knot happens in a matter of seconds, often without warning, and is catastrophic. The controlled experiments showed that increasing inertial effects of the swinging laces leads to increased rate of knot untying, that the directions of the impact and swing influence the rate of failure, and that the knot structure has a profound influence on a knot's tendency to untie under cyclic impact loading.

This is the story of my relationship with Patrick Dehornoy and his family, which began in 1996 and ended when Patrick passed away. [ABSTRACT FROM AUTHOR]

In this paper, we look at which Alexander and Markov theorems can be defined for generalized knot theories. [ABSTRACT FROM AUTHOR]

We construct a family of six dimensional block representations of the braid group B3 on three strings. We show that these representations can be used to distinguish braids of knots with 9 and 10 crossings from their reversed braids. [ABSTRACT FROM AUTHOR]

An experimental study was carried out to know the square mesh codend in trawl net by using different bridle lengths viz. 15 m and 20 m bridle length at a depth of 20 m. The study was carried out once in the month for a period of five months. The catch found was 8.7 kg and 13.5 kg for cover codend of 0day to the 15 m and 20 m bridle length respectively. As far as the 30th days are concerned, the catch obtained were 10.02 kg and 20.60 kg for 15 m and 20 m bridle length respectively. During the 60th day, the catch obtained was 12.02 kg and 30.50 kg for 15 m and 20 m bridle length respectively. At a depth of 20 m, the catch obtained for 90th day was 15.7 kg and 27.6 kg for the square mesh codend used with 15 m and 20 m bridle length respectively. Similarly for 120th day, the catch obtained was 14 kg and 25.5 kg for 15 m and 20 m bridle length respectively. During the study period, it was also observed that the diamond mesh was used as cover codend for the square mesh codend trawl net to know the escapement of fishes with 8.70 kg, 13.52 kg, 8.50 kg, 13.10 kg, 10.19 kg, 12.00 kg, 10.70 kg and 12.50 was observed for 0, 30th, 60th, 90th and 120th days respectively for the square mesh codend trawl net with a bridle length of 15 m and 20 m respectively. It was noticed that at a depth of 20 m, for a 20 m bridle length was found suitable for carrying the trawl net fishing operation commercially. [ABSTRACT FROM AUTHOR]

Copyright of Radio Electronics, Computer Science, Control is the property of Zaporizhzhia National Technical University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Out-of-grade pine timber is an abundant material resource that is underutilised because its mechanical properties are not well understood. Increasing trends toward shorter rotation times and fast-grown plantation pines around the world such as Pinus elliottii × P. caribaea var. hondurensis hybrid (PEE × PCH) mean low-stiffness corewood is becoming a larger portion of this out-of-grade population. This study characterised the modulus and strength properties in bending, compression parallel to grain (CParG) and compression perpendicular to grain (CPerpG), shear and tension strength of low-stiffness out-of-grade PEE × PCH. The effect of resin, knots and pith on these properties were also investigated. The results show that in clear wood, the MOE in bending, CParG, CPerpG and shear modulus are 6.9 GPa, 5.78 GPa, 0.27 GPa and 0.59 GPa, respectively, while strengths are 45.8 MPa, 29.4 MPa, 6.7 MPa, 5.7 MPa, respectively. The tensile strength is 32.4 MPa. Resin significantly increased density 45% higher than clear, but performed similar with the exception of CPerpG MOE and strength which were significantly different. Resin area ratio (RAR) has a moderate correlation with density with an R2 of 0.659 but low to no correlation for mechanical properties. Knots were significantly different to clear for all test types and within a range of 48% to 196%. Knots were high in CPerpG MOE and strength but lower for all other properties and had the largest negative impact on tensile strength. Knot area ratio (KAR) had low to moderate correlation with tension strength and CPerpG MOE with R2 of 0.48 and 0.35, respectively. Pith was within the range of 76% to 121% of non-pith samples for structural performance, some of which were significantly different, and pith samples were higher in density than non-pith. This new information is crucial for the effective establishment of grading rules, design optimisation and utilisation of low-stiffness out-of-grade PEE × PCH as a new material resource in civil engineering applications. [ABSTRACT FROM AUTHOR]

Families of 1‐ and 2‐periodic knots and weavings that have isogonal (vertex‐transitive) piecewise‐linear embeddings are described. In these structures there is just one thread, or multiple threads with parallel or collinear axes. The principal structures are a large family of 1‐periodic knots and related multi‐thread infinite links, knitting patterns and chain‐link weaving. The relevance to synthetic chemistry is described in terms of targets for designed synthesis such as mechanically interlocked polymers. [ABSTRACT FROM AUTHOR]

In this paper, we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing radii. Specifically, we extract features from the persistent homology (PH) of the Vietoris–Rips complexes built from point clouds associated with knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated with the embedding and PH-based features, as a function of the knots' lengths. We further study the variation of these correlations for different knot types. Finally, this framework also allows us to define a simple notion of deviation from ideal configurations of knots. [ABSTRACT FROM AUTHOR]

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot graphs. We also investigate the property of homogeneity, and prove that the concordance knot graph is homogeneous. Finally, we prove that that for any n, there exists a knot K such that the ball of radius n in the Gordian graph centered at K contains no connected sum of torus knots. [ABSTRACT FROM AUTHOR]

We present a broad selection of results on endomorphisms and automorphisms of the Cuntz algebras On that have been obtained in the last decades. A wide variety of open problems is also included.

This paper gives new upper bounds on the stick numbers of the knots 9 1 8 , 1 0 1 8 , 1 0 5 8 , 1 0 6 6 , 1 0 6 8 , 1 0 8 0 , 1 0 8 2 , 1 0 8 4 , 1 0 9 3 , 1 0 1 0 0 , and 1 0 1 5 2 , as well as on the equilateral stick number of 1 0 7 9 . These bounds imply that the knots 1 0 5 8 , 1 0 6 6 , and 1 0 8 0 have superbridge index ≤ 5 , 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. [ABSTRACT FROM AUTHOR]

We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide. [ABSTRACT FROM AUTHOR]

Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in three-space. In this manuscript, we examine the topological complexity of polygonal chains in three-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in three-space and by uniform random walks (URWs) in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For URWs in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as O (n 2) with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in three-space increases as O (n). These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement. [ABSTRACT FROM AUTHOR]

We introduce new polynomial invariants for both planar knotoids and spherical knotoids. To introduce these invariants, we first introduce what we call polar knots, and give the fundamentals of their theory. Polar knots are knots with a finite number of points that do not allow strands slide over or under them. We focus on polar knots with two poles, which we call 2-polar knots. We show that knotoids are in one-to-one correspondence with a subset of 2-polar knots. We obtain this correspondence through what we call the right traffic closure (or left traffic closure) of knotoids, which relates every knotoid with a 2-polar knot. We use Kauffman bracketology to define isotopy invariants for 2-polar knots and hence for knotoids. Our polynomial invariant of planar knotoids and planar 2-polar knots, denoted by X p , involves eight variables, and it can distinguish between a knotoid and its reverse unlike other polynomials previously defined using Kauffman bracket. Our invariant of spherical knotoids and spherical 2-polar knots, denoted by X s , involves four variables. We also define some geometric invariants of 2-polar knots and we conjecture that they can be detected by the X p and X s polynomials. [ABSTRACT FROM AUTHOR]

We study and characterize the topology of connectivity circuits observed in natively folded protein structures whose coordinates are deposited in the Protein Data Bank (PDB). Polypeptide chains of some proteins naturally fold into unique knotted configurations. Another kind of nontrivial topology of polypeptide chains is observed when, in addition to covalent bonds connecting consecutive amino acids in polypeptide chains, one also considers disulfide and ionic bonds between non-consecutive amino acids. Bonds between non-consecutive amino acids introduce bifurcation points into connectivity circuits defined by bonds between consecutive and nonconsecutive amino acids in analyzed proteins. Circuits with bifurcation points can form θ-curves with various topologies. We catalog here the observed topologies of θ-curves passing through bridges between consecutive and non-consecutive amino acids in studied proteins., (© 2024 The Protein Society.)

Flexural properties of visually graded southern pine structural lumber were evaluated. Several grade controlling characteristics were considered relative to bending properties and compared with current design values. A total of 751 southern pine lumber specimens were obtained from a broad spectrum of regions in the southeastern United States. Visually graded No. 2, nominally two inch thick specimens, in four and six inch widths, were obtained from commercial sawmills. All specimens were evaluated by a certified grader in the laboratory. Actual dimensions, weight, and moisture content (MC) were measured. Growth and manufacturing related characteristics were identified and classified into two categories: strength reducing characteristics (SRC) and grade reducing characteristics (GRC). Specific gravity (SG), bending modulus of elasticity (MOE), and modulus of rupture (MOR), were determined for each specimen. The presence of knots was identified as the most significant SRC; their presence had the most significant impact on SG, MOE and MOR. For GRC, specimens with knots, warp and specimens that fell into the category none, were significantly lower in SG, MOE and MOR. MOE and the allowable design bending strength values yielded in this study met the current design value criteria for both widths tested. [ABSTRACT FROM AUTHOR]

Crossarms are heavy timbers used to support electrical lines on utility poles. The requirements for crossarm material are fairly strict especially with respect to limitations on the presence of knots, which represent a gap or hole in the timber and also result in grain deviations that can affect properties. The strict grading rules can sometimes result in shortages of acceptable material. The potential for accepting arms with minor, but currently unacceptable defects was explored on a population of 200 rejected Douglas-fir crossarms and 50 arms that met current standards. Knot diameter and location on the arms were mapped and the arms were tested to failure in a specially designed apparatus that simulated an actual line load. The resulting data were used to calculate Modulus of Rupture (MOR) and Modulus of Elasticity (MOE). Nearly all arms failed above the minimum American National Standards assumed MOR value of 53.78 MPa and many of the rejected arms had higher MOR values than acceptable arms. While there was more variability in properties for the rejected arm population, correlations between knot diameter or knot location and MOR were all exceeding poor (r2 ≤ 0.25). The results suggest the need for further evaluation of the emphasis on knots in grading rules. [ABSTRACT FROM AUTHOR]

We prove that a minimal disc in a CAT.0/space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fáry-Milnor's theorem in the CAT.0/setting. [ABSTRACT FROM AUTHOR]