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This study is aimed to investigate the types of knot failure (untying or breaking) and the tension required to break different suture diameters. A total of 150 knots were fabricated using polyamide sutures with diameters of 6/0, 7/0, and 8/0. The studied knots were either squared or slipped with different numbers of throws (2, 3, 4, 5, and 6), and the following data were recorded: type of failure (untied or broken), number of throws, the tension required to untie or break each knot, slippage, and elongation of the knot. The knots were created in a standardized way with a device and weights, then subjected to a controlled tension. The knots that became untied were: 1=1, 1×1, 2=1, and 2×1, whereas the remaining knots broke. Notably, at least three throws were required to prevent untying, but separately, as in 1=1=1 or 1×1×1. The mean tension needed to break the knots in 6/0, 7/0, and 8/0 sutures were 3.1, 1.3, and 0.6 N, respectively (P < .05), and they were independent of the knot type. The results from this study demonstrate that the knots with geometries of 2=2/2×2 and 1=1=1/1×1×1 were secure, and having additional throws does not increase their security. Further, tensile strength reduces with decreased suture size. [ABSTRACT FROM AUTHOR]

This paper is the second installment of our in-depth study on the braid index problem of pretzel links. We classify pretzel links into three categories based on the Seifert circle decompositions in their standard link diagrams. In our initial paper, we successfully identified the braid indices for all Type 1 and Type 2 pretzel links. In this paper, we share our findings on Type 3 pretzel links. We are able to determine the exact braid indices for most Type 3 pretzel links. For the small subset of Type 3 pretzel links where exact braid indices remain undetermined, we establish their braid indices within a range of two consecutive integers. With some numerical evidence, we conjecture that in such an exceptional case, the braid index of the Type 3 pretzel link is given by the larger of the two consecutive integers given by our formulas. [ABSTRACT FROM AUTHOR]

In this paper and its sequel, we study the braid indices for all non-alternating pretzel links by a systematic approach. Pretzel links can be divided into three types according to the Seifert circle decompositions of their standard link diagrams. In this paper, we present our results on Type 1 and Type 2 pretzel links, namely the complete formulation of the braid indices for all non-alternating Type 1 and Type 2 pretzel links. Since the braid indices are already known for all alternating pretzel links from our previous work, it means that we have now completely determined the braid indices for all Type 1 and Type 2 pretzel links. [ABSTRACT FROM AUTHOR]

X-ray computed tomography (CT) is utilised in some sawmills today, primarily for enhancing value yield and for process automation, which includes log sorting and sawing optimisation. Nevertheless, there is a scarcity of recent research utilising CT to assess the local cutting process. As a preliminary study, this paper addresses this gap by using CT to investigate the connections between local cutting force and local wood properties including density, knots, and annual ring width. Workpieces of Scots pine (Pinus sylvestris L.), from Sweden and Poland, were CT-scanned in laboratory conditions. Quasi-linear cutting tests were then performed on both clear and knotty regions of the workpieces using a custom-made laboratory stand with a Stellite-tipped tooth mounted on piezoelectric sensors. It was found that density influences cutting forces for both clear and knotty wood, and this effect increased noticeably with increasing uncut chip thickness. Changes in wood density, such as between sapwood and heartwood or between clear wood and knot, caused dynamic changes in cutting forces and temporary disturbances to the stability of the system. Normalisation of cutting forces by local density allowed the conclusion that density is not the only property affecting cutting forces. Other structural properties, e.g. annual ring width and latewood–earlywood proportion may affect the cutting process as well, which requires deeper analysis in the future research. This preliminary study demonstrates the feasibility and usefulness of coupling CT data with cutting force measurements and suggests further research on the relationship between cutting force and wood properties. [ABSTRACT FROM AUTHOR]

We suggest a new random model for links based on meander diagrams and graphs. We then prove that trivial links appear with vanishing probability in this model, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained for a fixed number of crossings. A random meander diagram is obtained through matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of random links in this model, and, moreover, of the respective 3-manifolds that are link complements in 3-sphere. We use this for exploring geometric properties of a link complement. Specifically, we give expected twist number of a link diagram and use it to bound expected hyperbolic and simplicial volume of random links. The tools from combinatorics that we use include Catalan and Narayana numbers, and Zeilberger's algorithm. [ABSTRACT FROM AUTHOR]

Background Temporary pacemaker lead implantation is a common low-risk procedure, but can occasionally get complicated by infections, arrhythmias, thromboembolic events, and perforation of the vessel or the heart. However, intracardiac knotting of the temporary pacemaker lead has been rarely reported. This could lead to vascular or valvular injury, pneumothorax, symptomatic loss of pacing or haemodynamic compromise, and difficult lead removal. Case summary We are reporting a case of twice twice-knotted temporary pacemaker lead, which to our knowledge has not been reported before. The two knots in the transjugularly inserted temporary pacemaker lead, via a 6F venous sheath made it difficult to retrieve it. Discussion We decided to snare the knotted TPI into the inferior vena cava, and then retrieve it via a large-size femoral sheath, thus avoiding the need for a venotomy or any surgical intervention. [ABSTRACT FROM AUTHOR]

Stretchable devices have gained increasing interest in recent years, particularly in the field of wearable electronics. Among them, fiber‐type devices with high mechanical conformability hold great potential to enable next‐generation wearable and interactive applications with their special structure and high compatibility with the well‐established textile industries. In this study, a hydrogel fiber providing large moisture retention and high mechanical compliance is fabricated, with which a new approach to enable highly stretchable electromechanical sensors based on knot structures is developed. Comparative analysis with common orthogonal textile structures reveal the superior performance of sensors based on ionotronic knots. Stress sensors with the double overhand knot exhibit ≈four times greater variation in capacitance than those with orthogonal structures, and sensors with the clove hitch knot exhibit a fast response time of 57 ms. Based on the characteristics of different knots, a sensor matrix based on clove hitch knots to map the pressure distribution, and a wearable mole code generator based on reef knots to recognize and encode wrist motions are developed. These applications demonstrate the excellent performance of knot‐architecture sensors and their great potential in the fields of smart fabrics and human–machine interactions. [ABSTRACT FROM AUTHOR]

We give a homotopy classification of the global defects in ordered media and explain it via the example of biaxial nematic liquid crystals, that is, systems where the order parameter space is the quotient of the three-sphere S3 by the quaternion group Q. As our mathematical model, we consider continuous maps from complements of spatial graphs to the space S3/Q modulo a certain equivalence relation and find that the equivalence classes are enumerated by the six subgroups of Q. Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of Q ; once we pass to planar diagrams, these labels can be refined to elements of Q associated with each arc. The same classification scheme applies not only in the case of Q but also to arbitrary groups. [ABSTRACT FROM AUTHOR]

The effects of wood temperature, cutting width, and knots on the surface quality of black spruce cants processed by a chipper-canter were evaluated. Four matched groups of logs were machined at temperatures of 20°C, 0°C, −10°C, and −20°C. Each log was transformed at two cutting widths (CW: 12.7 and 25.4 mm). Knot characteristics were measured on the cant surfaces after log processing. Surface quality of cants was assessed by roughness and waviness parameters and torn grain. The quality of surfaces was affected by the temperature of logs and cutting width. Poorer surface quality was obtained at larger cutting widths. This was caused by increased cutting forces when processing more material at larger cutting widths, compounded by the presence of knots. Waviness and roughness were higher for frozen logs than for unfrozen logs. Although the sub-zero temperatures caused higher cutting forces and vibrations, their effect was partly offset by the strengthening of the earlywood and the brittle behaviour of frozen wood. Correlations and regression analyses showed that the optimisation of the cutting conditions for decreasing waviness and roughness should also reduce the torn grain depth. Moreover, the position and area of knots could be considered to minimise waviness and roughness and the occurrence of torn grain. [ABSTRACT FROM AUTHOR]

We present a family M g of fibered hyperbolic 3-manifolds whose fiber F g is the unique connected incompressible surface and the genus g ≥ 2 of F g can be arbitrary. This answers a question of Agol. [ABSTRACT FROM AUTHOR]

Escape the Math Room is a fun, interactive, sequence of puzzles designed to encourage mathematical collaboration, team work, and ingenuity. Using advanced mathematical topics in an accessible way, Escape the Math Room appropriately challenges students ranging from high school to advanced undergraduate. The activity can be adjusted by adding or removing hints and explanations to accommodate the audience. This activity can be used on its own for a mathematical event, in class on a "free day" or as a team building exercise. [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. [ABSTRACT FROM AUTHOR]

Matrix-valued holomorphic quantum modular forms are intricate objects associated to 3-manifolds (in particular to knot complements) that arise in successive refinements of the volume conjecture of knots and involve three holomorphic, asymptotic and arithmetic realizations. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the (- 2 , 3 , 7) -pretzel knot. [ABSTRACT FROM AUTHOR]

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs. [ABSTRACT FROM AUTHOR]

A category equivalent to the category of 3-dimensional cobordisms is defined in terms of planar diagrams. The operation of composition in this category is completely described via these diagrams. [ABSTRACT FROM AUTHOR]

Introduction. Lateral ankle sprains are the most common type of injury to the ankle and can lead to ankle instability. There are many described techniques for the surgical treatment of lateral ankle instability. The purpose of this study is to quantify the variation in surgeon technique for lateral ankle instability treatment. Methods. Surveys were sent to 62 orthopaedic foot and ankle surgeons regarding surgical technique for the treatment of lateral ankle instability. Clinical agreement was defined as greater than 80% agreement to assess the cohesiveness of surgical methods as described by Marx et al. Results. Response rate was 49/62 (79%). There was clinical agreement for not using bone tunnels and not using metal anchors. All other factors lacked clinical agreement. A greater average number of throws and knots (4.2 for each, range 1-6 throws, range 2-12 knots) were used by surgeons that do not believe knots cause pain compared to an average of 3.9 (range, 1-6) throws and 4.0 (range, 2-15) knots by surgeons who do believe knots cause pain. The association that surgeon who believed knots do cause pain and thus used fewer knots and throws was not statistically significant (P >.05). The preferred material by surgeons in our study are as follows: nonabsorbable braided suture (26/49, 53%), suture tape (15/49, 31%), and fiber tape (4/49, 8%). Among surgeons who use absorbable suture (34/49, 69%), there was no significant difference (P >.05) between surgeons who believe knots cause pain (23/34, 68%) and those who do not (11/34, 32%). Discussion and Conclusion. Among this small sample of orthopaedic foot and ankle surgeons, there is wide variation in surgical technique for lateral ankle instability treatment and little agreement on the clinical standard of care. This disagreement highlights the need for comparative outcome studies in the treatment of ankle instability. Level of Evidence: Level III: Retrospective cohort study [ABSTRACT FROM AUTHOR]

Deformable linear objects (DLOs), such as rods, cables, and ropes, play important roles in daily life. However, manipulation of DLOs is challenging as large geometrically nonlinear deformations may occur during the manipulation process. This problem is made even more difficult as the different deformation modes (e.g., stretching, bending, and twisting) may result in elastic instabilities during manipulation. In this paper, we formulate a physics-guided data-driven method to solve a challenging manipulation task—accurately deploying a DLO (an elastic rod) onto a rigid substrate along various prescribed patterns. Our framework combines machine learning, scaling analysis, and physical simulations to develop a physics-based neural controller for deployment. We explore the complex interplay between the gravitational and elastic energies of the manipulated DLO and obtain a control method for DLO deployment that is robust against friction and material properties. Out of the numerous geometrical and material properties of the rod and substrate, we show that only three non-dimensional parameters are needed to describe the deployment process with physical analysis. Therefore, the essence of the controlling law for the manipulation task can be constructed with a low-dimensional model, drastically increasing the computation speed. The effectiveness of our optimal control scheme is shown through a comprehensive robotic case study comparing against a heuristic control method for deploying rods for a wide variety of patterns. In addition to this, we also showcase the practicality of our control scheme by having a robot accomplish challenging high-level tasks such as mimicking human handwriting, cable placement, and tying knots. [ABSTRACT FROM AUTHOR]

We introduce a quandle version of the normalized (twisted) Alexander polynomial, which is an invariant of a pair of an oriented link and a quandle representation. The invariant can be constructed by fixing each Alexander pair, and we find various invariants in our framework, which include the quandle cocycle invariant and the normalized (twisted) Alexander polynomial of a knot. In this paper, we develop the theory of normalization with row and column relations of matrices. The theory works for several row and column relations, although the twisted Alexander polynomial is defined with one column relation. We give a formula of our invariant for the mirror image of an oriented link, which explains why the Alexander polynomial fails to detect the chirality of knots and why the quandle cocycle invariant effectively detects it from a unified point of view. We also show that cohomologous Alexander pairs yield the same invariant. [ABSTRACT FROM AUTHOR]

The fisherman's knot, renowned for its strength and reliability, finds applications in engineering and medicine. However, a comprehensive understanding of its mechanics remains limited in scientific literature. In this paper, we present a systematic study of the tightening behavior of the fisherman's knot through a combined approach of tabletop experiments and discrete elastic rods simulations. Our experimental setup involves gradually applying tension to the two ends of the fisherman's knot until it fractures. We observed a correlation between the knot's material properties and its behavior during tightening, leading up to fracture. The tightening process of the fisherman's knot exhibits distinct "sliding" or "stretching" motions, influenced by factors such as friction and elastic stiffness. Furthermore, the failure modes of the knot (material fracture and topological failure) are determined by an interplay between elastic stiffness, friction, and initial conditions. This study sheds light on the underlying mechanics of the fisherman's knot and provides insight into its behavior during the tightening process, contributing to the broader understanding of the mechanics of knots in practical applications. [ABSTRACT FROM AUTHOR]

The use of 2,5‐diformylpyrrole in self‐assembly reactions with diamines and Zn(II)/Cd(II) salts allowed the preparation of [2]catenane, trefoil knot, and Borromean rings. The intrinsically dynamic nature of the diiminopyrrole motif rendered all of the formed assemblies intramolecularly flexible. The presence of diiminopyrrole revealed new coordination motifs and influenced the host–guest chemistry of the systems, as illustrated by hexafluorophosphate encapsulation by Borromean rings. [ABSTRACT FROM AUTHOR]

This essay has two sections. The first explores the origin of the universe, the four elements, the knot of love, the third essence, and 'Primordial Vessel' by the Florentine Neoplatonic philosopher Marsilio Ficino. The second part interprets Botticelli's The Birth of Venus of 1485–1486, now at the Galleria degli Uffizi in Florence, drawing upon Ficino's Neoplatonic texts. Particularly some of Ficino's Commentaries on Plato's Dialogues (published in 1484), Symposium (completed in 1469), Philebus (completed in 1469), and Compendium on Timaeus (1480–1491). To this is added Ficino's Theologia Platonica (published in 1482), his Letters (published in 1495), his Commentary on Proclus's Alcibiades I (completed in 1488), and his translation of the Neoplatonic text Corpus Hermeticum , entitled Liber Mercurii Trismegisti de potestate et sapientia Dei (published in 1471). Botticelli's beautiful painting captures the depiction of the goddess Venus's birth as described in Hesiod's Theogony , which Ficino recounted in his Philebus Commentary [ABSTRACT FROM AUTHOR]

In this study, we develop a perspective on the diverse aesthetics historically associated with mathematics, inspired by Rancière’s approach to aesthetics and politics. We call “Silencing Aesthetics” a dominant aesthetic that Rota has characterized as a “copout (...) intended to keep our formal description of mathematics as close as possible to the description of a mechanism”. The challenge this study attempts to explore is how to question silencing aesthetics to make space for inclusive ones. Our efforts have focused on setting up and studying inclusive and pluralist “Studios”, gathering craftworkers, anthropologists, mathematics educators, and mathematics enthusiasts. We include here a case study based on a conversation amongst basket weavers, anthropologists, and mathematics educators focused on the artisanal and mathematical nature of knots. We discuss the implications of aesthetical entanglements, such as those in our case study, for mathematics learning. [ABSTRACT FROM AUTHOR]

We studied the healing strength and histological changes of digital flexor tendons repaired using Kessler (core suture knots placed over the tendon surface) and modified Kessler (core suture knots placed between two tendon ends) in 31 long toes of chicken. Four weeks after surgery, the healing tendons were measured in a tensile testing machine, and the adhesion formation and histological changes were observed. The strength of the Kessler repairs was significantly greater than that of the modified Kessler repairs with a 35% mean difference. No significant difference was found between the adhesion scores of the tendons repaired with both techniques. In histological sections, the arrangement of collagen fibers in the modified Kessler repair group was more disordered. We conclude that the tendons repaired with the Kessler method are stronger than those with the modified Kessler technique. The knots between tendon ends are detrimental to the early healing strength of digital flexor tendons. [ABSTRACT FROM AUTHOR]

The Neumann–Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the construction of a number of quantum invariants that include the loop invariants, the 3D-index and state-integrals. We define a twisted version of Neumann–Zagier matrices, describe their symplectic properties, and show how to compute them from the combinatorics of an ideal triangulation. As a sample application, we use them to define a twisted version of the 1-loop invariant (a topological invariant) which determines the 1-loop invariant of the cyclic covers of a hyperbolic knot complement, and conjecturally is equal to the adjoint twisted Alexander polynomial. [ABSTRACT FROM AUTHOR]

In this work, we find a closed form formula for the braid index of an n -bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary, effective, and self-contained, and partially recovers work of Birman–Kofman. Along the way, we show that the disparate definitions of twisted torus knots in the literature agree. [ABSTRACT FROM AUTHOR]

In this study, we develop a perspective on the diverse aesthetics historically associated with mathematics, inspired by Rancière's approach to aesthetics and politics. We call “Silencing Aesthetics” a dominant aesthetic that Rota has characterized as a “copout (…) intended to keep our formal description of mathematics as close as possible to the description of a mechanism”. The challenge this study attempts to explore is how to question silencing aesthetics to make space for inclusive ones. Our efforts have focused on setting up and studying inclusive and pluralist “Studios”, gathering craftworkers, anthropologists, mathematics educators, and mathematics enthusiasts. We include here a case study based on a conversation amongst basket weavers, anthropologists, and mathematics educators focused on the artisanal and mathematical nature of knots. We discuss the implications of aesthetical entanglements, such as those in our case study, for mathematics learning.

In this paper, we introduce a new polynomial invariant of planar (but not spherical) knotoids, which we call the winding signed sum polynomial. This Laurent polynomial invariant of planar knotoids denoted by β K is a type-one Vassiliev invariant. This invariant might tell whether a planar knotoid is a zero-height or nonzero-height planar knotoid. It also might distinguish between a planar knotoid and its inverse, while the affine index polynomial defined by Gügümcü and Kauffman in [New invariants of knotoids, Eur. J. Combin.65 (2017) 186–229] (that is also a type-one Vassiliev invariant) cannot distinguish between a planar knotoid and its inverse. We also define some geometric invariants of planar knotoids, and give lower bounds for these invariants using the winding signed sum polynomial, which helps in computing these geometric invariants that are easy to define, but hard to calculate. [ABSTRACT FROM AUTHOR]

We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construction works under the assumption that the algebra is braided commutative. The resulting knot invariant is a module with a coadjoint action. Taking the coinvariants yields a new quantum character variety that may be thought of as an alternative to the skein module. We give concrete examples for a few of the simplest knots and links. [ABSTRACT FROM AUTHOR]

Knot theory is a distinct branch of topology. In recent years, exciting new application of knot theory to biology and chemistry, especially to DNA structures are being developed. Alexander polynomial is very closely connected with the topological properties of knots. In this paper we have determined the solution of Ak and & knots using the condition of Alexander polynomial and Jones polynomial. [ABSTRACT FROM AUTHOR]

The evolution toward small-diameter and fast-growing plantation timbers such as the Pinus elliotti var. elliottii (Engelm) × Pinus caribaea var. hondurensis (Sénéclauze) (PEE×PCH) hybrids around the world is producing large volumes of core wood that are falling short of structural sawn timber grading requirements. Engineered timber products such as cross-laminated timber (CLT) and glue-laminated (glulam) offer potential solutions to value-adding this resource, but the bond performance of this feedstock and the extent to which current standards and guides address its common characteristics for bond performance need to be understood. This study investigated the bond quality and performance of clear defect-free, low stiffness out-of-grade PEE×PCH and evaluated this performance using the pass/fail criteria of the CLT bond performance requirements of three national CLT standards. 5-layer CLT delamination samples and shear block test samples were glued using one-component polyurethane (PUR). This process was repeated for common occurring characteristics in this resource of resin, knots, and pith to understand their impact and inform an evaluation on the need to restrict their inclusion. Clear samples had an average glue line delamination of 2.9% and an average glue line wood failure of 96.7%. Resin achieved 9.3% and 92.6%, respectively. While knots had the lowest performance at 24.4% and 77.4%, respectively. When pith was at or adjacent to the glue line, wood failure occurred through the pith and its immediate surrounding fiber. Shear strength and wood failure tests were carried out on glulam and CLT-oriented samples. CLT knot samples were tested in two load orientations. Glulam-oriented samples in clear, resin, pith, and knots achieved an average shear strength of 8.5 MPa, 8.2 MPa, 7.9 MPa, and 8.2 MPa, respectively, and wood failure of 86%, 85%, 90%, and 69%, respectively. CLT-oriented samples in clear and resin both achieved average shear strengths of 4.0 MPa; 0°-loaded and 90°-loaded pith samples achieved 3.6 MPa and 2.4 MPa, while 0°-loaded and 90°-loaded knot samples achieved 4.2 MPa and 4.7 MPa respectively. Average wood failures were 90%, 89%, 96%, 96%, 83%, and 51%, respectively. PRG320 was found to be the most restrictive standard. Resin, knots, and pith were not addressed in the evaluation of delamination or shear strength in any standard, and PRG320 was the only standard to restrict these characteristics over and above structural grading rules. The amount and type of characteristics present vary considerably in structurally graded wood, and even more so for this out-of-grade resource. It was determined that the negative impact that resin, knots, and pith have on bond quality and bond performance calls for some restriction of their inclusion in order to achieve the author's interpretation of the intended bond performance requirements of the CLT standards, which currently do not address these characteristics well or at all. A proposed modification to the PRG320 effective bond area was presented as a proactive solution. [ABSTRACT FROM AUTHOR]

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a boundary-parabolic PSL (2 , ℂ) -representation to boundary-unipotent SL (2 , ℂ) -representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group. [ABSTRACT FROM AUTHOR]

In this work, we reflect on a speculative pedagogical exercise in which we recorded, and invited others to record, through and with knots, situations of our daily lives related to forms of containment and entrapment. Accompanying this creative and collaborative work allowed us to materialize situations of vulnerability and unease (as knots), as well as the difficulty of registering these discomforts (through knots). We are interested in presenting this exercise as an embodied design practice. We also want to highlight how the staging of vulnerability and unease―made through material practices― generates contemplative pauses that have the power to make the fragility of life visible, and thus anticipate the need for repair. We argue here that repair, as a practice linked to design, is a making that deserves to be propitiated by contemplation and catharsis, in the face of damage and fragility in capitalist contexts. [ABSTRACT FROM AUTHOR]

This study presents opportunities for the utilization of timber by-products with defects for manufacturing engineered wood panels. Three gluing methods were proposed for this waste raw material derived from Scots pine (Pinus sylvestris L.) wood. The methods used for combining and gluing enabled a more complete and complex utilization of wood with defects. The physical properties (density and moisture content) and mechanical properties (bending strength and modulus of elasticity) of the laboratory-fabricated engineered wood panels were evaluated in accordance with the European standards. The highest density of 643 kg/m3 and bending strength values (28.6 N/mm2) were obtained from the panels manufactured using method 3 and veneered with beech veneer sheets. The modulus of elasticity of the laboratory-made engineered wood panels reached values of up to 5580 N/mm2. This study demonstrated the feasibility of the utilization of defective wood pieces in the manufacturing of engineered wood panels.

We introduce new topological quantum invariants, surface link quantum invariants, of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed by using virtual knots and links.These invariants are new, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary.Our new invariants, surface link quantum invariants, give new invariants of classical knots and links, i.e., knots and links in the 3-sphere. That is, virtual knot theory induces a new, classical knot invariant. [ABSTRACT FROM AUTHOR]

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in R 4 is a topological knot (or link) in S 3 . We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in R 4 and knot theory. Namely, for any knot K, we construct a surface X K in R 4 such that: the link at the origin of X K is a trivial knot; the germs X K are outer bi-Lipschitz equivalent for all K; two germs X K and X K ′ are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and K ′ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in R 4 , even when they are topologically trivial and outer bi-Lipschitz equivalent. [ABSTRACT FROM AUTHOR]

We review a constructions of knots from elements of the Thompson groups due to Vaughan Jones, which comes in two flavors: oriented and unoriented. [ABSTRACT FROM AUTHOR]

The partition function of complex Chern–Simons theory on a 3-manifold with torus boundary reduces to a finite-dimensional state-integral which is a holomorphic function of a complexified Planck's constant τ in the complex cut plane and an entire function of a complex parameter u. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear q-difference equation. We further conjecture that a distinguished entry of the Stokes automorphism matrix is the 3D-index of Dimofte–Gaiotto–Gukov. We provide proofs of our statements regarding the q-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic 4 1 and 5 2 knots. [ABSTRACT FROM AUTHOR]

Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful "gliding ideas" in the literature — sometimes in disguise — such as the Drinfel'd double construction, Enriquez's work on quantization of Lie bialgebras, and Audoux and Meilhan's classification of welded homotopy links. [ABSTRACT FROM AUTHOR]

We deal here with three metastable systems: 1° – the dowser texture, 2° – supertwisted cholesterics and 3° – hypotwisted cholesterics. We outline remarkable properties of the tropisms of the dowser texture stemming from its low symmetry, and we show that, using setups called Dowsons Colliders, nematic monopoles can be nucleated, set into motion and brought into collisions in the dowser texture. Subsequently, we point out that nucleation of dislocation loops occurs in cholesteric layers compressed or dilated between cylindrical mica sheets. Under compression, three modes of nucleation of dislocation loops have been identified: individual, serial and continuous. The serial mode generates dislocation nets made of L circular loops connected by C radial crossing into superloops. Similar superloops can also be generated under dilation. We analyse the topology of superloops in terms of the theory of knots and point out that they can be reduced to the unknot by Reidemeister moves. In other words, superloops are multiply folded loops that can be continuously unfolded. [ABSTRACT FROM AUTHOR]

Objective: The primary aim of this study was to explore the relationship between the biophysical structure and function of modern suture materials. Particularly the suture's ability to withstand the stressors of surgery and how the material properties affect knot stability. The secondary aim was to investigate the effect that different knots have on the suture material itself. This study builds on previous research assessing suture and knot characteristics but in modern Ultra High Molecular Weight Polyethylene (UHMWPE) materials currently in widespread clinical use in arthroscopic surgery. Methods: Three common UHMWPE sutures and one polyester suture were tested in both a dry and wet state using the Geelong, Nicky's, Surgeon's and Tautline knots. Tensile strength of knots was tested vertically at a 60 mm/min strain rate and 45 mm gauge length. Sutures were tied through a cannula around two 8 mm diameter circular bollards. Testing was conducted in a controlled environment temperature and humidity environment (20 ± 2 °C, 65 ± 2%). Results: No one knot type was optimal over all suture types. Mean tensile strength in both a dry and wet state and a low coefficient of variation (CV) in tensile strength in a wet state were considered as an indication of suitability. With Ethibond sutures this was the Geelong knot (CV:4.2%). With Orthocord sutures both the Geelong and Tautline knots (CV:4.2% and CV:11.9% respectively). With FiberWire sutures the Nickys and Tautline knots (CV:22.6% and CV:22.5% respectively). With ForceFiber sutures all four knots exhibited similar wet tensile strength with high variability showing that all should perform in a similar way invivo. Conclusions: This study demonstrates a statistically significant three-way interaction between polyblend suture materials, the knot and the environment. This has implications for knot security using the tested sutures in different environments, as one knot may not behave the same under all conditions.

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]

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]