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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]

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]

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]

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]

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]

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]

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]

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 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 consider -difference equations for colored Jones polynomials. These sequences of polynomials are invariants for the knots and their asymptotics plays an important role in the famous volume conjecture for the complement of the knot to the d sphere. We give an introduction to the theory of hyperbolic volume of the knots complements and study the asymptotics of the solutions of -recurrence relations of high order. [ABSTRACT FROM AUTHOR]

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of S 3 branched along a knot α ⊂ S 3 . Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. [ABSTRACT FROM AUTHOR]

John Conway was one of the most versatile mathematicians in modern times, who made important contributions to several areas of mathematics. In this article, we highlight his contributions to two areas—knot theory and group theory. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov–Bohm solenoids in the Euclidean three-space, the flux carried by an oriented knot features a 2 π -periodicity of the associated operator. For a given link, one thus obtains a family of Dirac operators indexed by a torus of fluxes. We study the spectral flow of paths of such operators corresponding to loops in this torus. The spectral flow is in general nontrivial. In the special case of a link of unknots, we derive an explicit formula for the spectral flow of any loop on the torus of fluxes. It is given in terms of the linking numbers of the knots and their writhes. [ABSTRACT FROM AUTHOR]

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups F → and T → of the Thompson groups F and T, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones–Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups F → and T → . Within this framework, we discuss an alternative approach that relies on some partition function interpretation of the Jones polynomial, and also exhibit more examples associated with other link invariants, notably the two-variable Kauffman polynomial and the HOMFLY polynomial. In the unoriented case, extending our previous results, we also show by similar methods that certain evaluations of the Tutte polynomial and of the Kauffman bracket, suitably renormalized, yield functions of positive type on T. [ABSTRACT FROM AUTHOR]

This article is intended to give the reader a flavor of three-dimensional topology in an informal setting. [ABSTRACT FROM AUTHOR]

Increasing the objectivity of comparison and ranking of working efficiency of steam-turbine installations of thermal power-stations' power-generating units under conditions when the accuracy of technical and economic indices (TEI) to be calculated by energy characteristics leaves much to be desired; in other words, the results of TEI analysis are far from objective, and the improvement of analysis techniques of TEI is not only actual but also necessary. Traditionally, the analysis consists of comparison of actual and normal values of TIEs. The discrepancies of these values define the reserve of thermal economy and recommendations ensuring the enhancement of working efficiency of thermal power stations (TPS). Systematic improvement of recommendations' reliability is ensured by experimental clarification of energy characteristics 4 years on average after the major repair of a power-generating unit (PGU). In time, wear leads to increasing the calculation error of rated values of techno-economic indices and amendments taking into account aging. Finally, there comes a point when a manual clarification of rated TEIs becomes economically unsuitable. In accordance with the Rules for Technical Operation, the analysis results of TEIs are needed to ensure the most economical operation of powergenerating units, the development of schedules of repair and adjustment and alignment of power-generating units, the estimation of quality of operational personnel work, and drawing up of accounts on the thermal efficiency. Personnel should solve all these problems, as well as many others, depending on the difference of the actual average monthly and approximate rated values of dozens of TEI. It is proposed, along with the methods of analysis, to apply the methods of synthesis of TEI. The methods of synthesis require overcoming the influence of difference in dimensions, the influence of interrelation of TEI, and the operating conditions of powergenerating units. The possibility for synthesis of TEI manually is practically excluded, since their bulkiness and multiplicity cause a high risk of erroneous decisions. An automated analysis and synthesis system of TEI of thermal power stations' power-generating units was developed. It provides, along with the information support of personnel, methodological support in the form of recommendations for enhancement of working effectiveness of both individual power-generating units and thermal power stations as a whole. The objectivity of these recommendations is undoubted within the framework of initial data. [ABSTRACT FROM AUTHOR]

In the paper [1], V. O. Manturov introduced the groups Gkn depending on two natural parameters n > k and naturally related to topology and to the theory of dynamical systems. The group G2n, which is the simplest part of Gkn, is isomorphic to the group of pure free braids on n strands. In the present paper, we study the groups G2n supplied with additional structures-parity and points; these groups are denoted by G2n,p and G2n,d. First,we define the groups G2n,p and G2n,d, then study the relationship between the groups G2n, G2n,p, and G2n,d. Finally, we give an example of a braid on n + 1 strands, which is not the trivial braid on n + 1 strands, by using a braid on n strands with parity. After that, the author discusses links in Sg × S1 that can determine diagrams with points; these points correspond to the factor S1 in the product Sg × S1. [ABSTRACT FROM AUTHOR]

The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. The author focuses on the case of G = SL( N, C) and M being a knot complement: M = S \ K. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G = SL( N, C). He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory. [ABSTRACT FROM AUTHOR]

Call a smooth knot (or smooth link) in the unit sphere in $$\mathbb {C}^2$$ analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let K be a smoothly analytic knot. For a small tubular neighbourhood of K we give a sharp lower bound for the 4-ball genus of analytic links L contained in it. [ABSTRACT FROM AUTHOR]

We give an excluded minor characterisation of the class of ribbon graphs that admit partial duals of Euler genus at most one. [ABSTRACT FROM AUTHOR]

Rossler positive braids are defined and denoted by R m + . We represented a general form of Rossler positive braid, and its associated permutation. For the m period Rossler positive braid R m + , the number of crossing equals to [ 2 ( m - 2 ) + ( m - 1 ) ] , and the genus g ( R m + ) = ( m - 2 ) , it also has a braid index equals to m , m > 1 . [ABSTRACT FROM AUTHOR]

The hanging mark is the most relevant feature of hanging and its characteristics are well known in the literature. Most of the time, the ligature material is not available during autopsy examination in hanging. Hence, the features of the ligature material are not submitted to systematic analysis. However, the type and position of the knot plays an important role in the mechanism of death and autopsy findings in hanging. Out of the total hanging deaths, complete hanging was seen in 67.7% of the cases, but a typical hanging was noted in only 10.2% of the cases. The commonest type of ligature material used for ligation around the neck was nylon rope followed by odhni and jute rope. The fixed knot was noted in 64.6% of the cases and a running (slip) in 21.3% of the cases. The commonest position of the knot was at nape of the neck, followed by the left side of the neck at mastoid process. The number of turns/loops of a ligature around the neck was one turn in 72.4% and two in 25.2% the hanging deaths. Most victims committed suicide by hanging in their homes, and the commonest ligature points were trees, flowed by beams and ceiling hook/fans. [ABSTRACT FROM AUTHOR]

A smooth simultaneous confidence band (SCB) is obtained for heteroscedastic variance function in nonparametric regression by applying spline regression to the conditional mean function followed by Nadaraya-Waston estimation using the squared residuals. The variance estimator is uniformly oracally efficient, that is, it is as efficient as, up to order less than $$n^{-1/2}$$ , the infeasible kernel estimator when the conditional mean function is known, uniformly over the data range. Simulation experiments provide strong evidence that confirms the asymptotic theory while the computing is extremely fast. The proposed SCB has been applied to test for heteroscedasticity in the well-known motorcycle data and Old Faithful geyser data with different conclusions. [ABSTRACT FROM AUTHOR]

We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that the non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Dehornoy ordering of the braid groups. [ABSTRACT FROM AUTHOR]

Witten's approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys-Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any G2 holonomy manifold and is a close cousin of the monopole equation of Bogomolny. Octonions play the central role in our view, in which both the seven-dimensional equations and the Haydys-Witten equations appear as reductions of the eight-dimensional Spin(7) instanton equation. [ABSTRACT FROM AUTHOR]

We address here the topological equivalence of knots through the so-called Reidemeister moves. These topology-conserving manipulations are recast into dynamical rules on the crossings of knot diagrams. This is presented in terms of a simple graphical representation related to the Gauss code of knots. Drawing on techniques for reaction-diffusion systems, we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded. The aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation. This necessitates the introduction of composite paulionic operators. The formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings, subject to topology-conserving stochastic dynamics. We consider here the simplified situation of two-dimensional knot projections. However, we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes. Further applicability arises in the context of the simulated annealing of knots. The methods presented here depart significantly from the invariant-based path integral descriptions often applied in polymer systems, and, in our view, offer a fresh perspective on the conservation of topological states and topological equivalence in knots. [ABSTRACT FROM AUTHOR]

Recent advances in Quantum Topology assign $$q$$ -series to knots in at least three different ways. The $$q$$ -series are given by generalized Nahm sums (i.e., special $$q$$ -hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those $$q$$ -series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges drawing several conclusions. In addition, we give a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of $$q^k$$ in those series for $$k=0,1,2$$ in terms of polynomials on the number of vertices, edges, and triangles of the graph. [ABSTRACT FROM AUTHOR]

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in $${\mathbb{R}^{3}}$$ . More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in $${\mathbb{R}^{3}}$$ , we show that they can be transformed with a C-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin. [ABSTRACT FROM AUTHOR]

We consider a varying coefficient regression model for sparse functional data, with time varying response variable depending linearly on some time-independent covariates with coefficients as functions of time-dependent covariates. Based on spline smoothing, we propose data-driven simultaneous confidence corridors for the coefficient functions with asymptotically correct confidence level. Such confidence corridors are useful benchmarks for statistical inference on the global shapes of coefficient functions under any hypotheses. Simulation experiments corroborate with the theoretical results. An example in CD4/HIV study is used to illustrate how inference is made with computable p values on the effects of smoking, pre-infection CD4 cell percentage and age on the CD4 cell percentage of HIV infected patients under treatment. [ABSTRACT FROM AUTHOR]

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram. Until recently, explicit formulas of this type were known only for few invariants of low degrees. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. We then identify the resulting invariants with certain derivatives of the HOMFLYPT polynomial. [ABSTRACT FROM AUTHOR]

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S. The key role is played by the $${SL(2, \mathbb{Z})}$$ transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Mariño. Compared to the recent proposal by Aganagic and Vafa for knots on S, we demonstrate that the disk amplitude of the A-brane associated with any knot is sufficient to reconstruct the entire B-model spectral curve. Finally, the construction of toric Lagrangian A-branes is generalized to other local toric Calabi-Yau geometries, which paves the road to study knots in other three-manifolds such as lens spaces. [ABSTRACT FROM AUTHOR]

We apply kneading theory to describe the knots and links generated by the iteration of renormalizable nonautonomous dynamical systems with reducible kneading invariants, in terms of the links corresponding to each factor. As a consequence we obtain explicit formulas for the genus for this kind of knots and links. [ABSTRACT FROM AUTHOR]

In this paper, we construct mosaic representations of knots on the torus, rather than in the plane. This consists of a particular choice of the ambient group $$\mathbb{A }$$, as well as different Definitions of contiguous and suitably connected. We present conditions under which mosaic numbers might decrease by this projection, and present a tool (called waste) to measure this reduction. We show that the order of edge identification in construction of the torus sometimes yields different resultant knots from a given mosaic when reversed. Additionally, in the Appendix we give the catalog of all torus $$2$$-mosaics. [ABSTRACT FROM AUTHOR]

Practical modeling of spatial surfaces is more convenient by means of transformation of their flat developments made as topologically connected kinetic structures. Topologically, any surface in 3D space consists of three types of elements: planar facets, linear edges and point vertexes. Planar facets and linear edges can be identifed respectively with the structural units of folding structures and kinematical nets. Here we consider a third possible type of flat transformable structures with vertexes as form-generative units, in which flat developments of surfaces are formed by arranged point sets given by contacting crossing points of the periodic knots and links made of elastic-flexible materials, so that their crossing points have real physical contacts. This new form-generative method can be applied to modeling of both oriented and non-oriented differentiable topological 2D manifolds. The method of form generation based upon the developing properties of periodic structures of knots and links may be applied to many practical fields, including art, design and architecture. [ABSTRACT FROM AUTHOR]

The amount and type of self-entanglement of DNA filaments is significantly affected by spatial confinement, which is ubiquitous in biological systems. Motivated by recent advancements in single DNA molecule experiments based on nanofluidic devices and by the introduction of algorithms capable of detecting knots in open chains, we investigate numerically the entanglement of linear, open DNA chains confined inside nano-slits. The results regard the abundance, type, and length of occurring knots and are compared with recent findings for DNA inside nano-channels. In both cases, the width of the confining region, D, spans the 30 nm-1 μm range and the confined DNA chains are 1-4 μm long. It is found that the knotting probability is maximum for slit widths in the 70-100 nm range. However, over the considered DNA contour lengths, the maximum incidence of knots remains below 20%, while for channel confinement it tops 50%. Further differences of the entanglement are seen for the average contour length of the knotted region, which drops significantly below D ~100 nm for channel-confinement, while it stays approximately constant for slit-like confinement. These properties ought to reverberate in different kinetic properties of linear DNA depending on confinement and could be detectable experimentally or exploitable in nano-technological applications. [ABSTRACT FROM AUTHOR]

Experimental laboratory study was made to test the knot integrity of identical, non-identical and parallel sliding knots, with three and five throws, made with synthetic resorbable multifilament suture materials. The knots were made with Novosyn (polyglactin 612), Safil (polyglycolic acid), Vicryl (polyglactin 612) and Vicryl plus (polyglactin 910 + triclosan), all with suture size: 3-0 USP. Per material 10 knots for every kind of knot were tested in a tensiometer, resulting in a total of 240 tests. Sliding knots with three throws were compared with the five throw sliding knots, and a comparison of the loop-holding capacities (LHC) of the different suture materials was made. Differences in suture material, knot type, and number of throws in the knot had a remarkable effect on knot performance. Adding two extra throws to a three throw non-identical or parallel sliding knot resulted in significantly more reliable knots ( P < 0.05). In identical sliding knots, this effect was not seen, but these knots showed low LHCs, indicating poor knot reliability. Compared to the other suture materials, Safil showed significantly lower LHCs. Most of the mean LHCs of the various knots with Vicryl, Vicryl Plus or Novosyn were not statistically different from each other. Identical sliding knots appeared to be very unreliable, especially when made with three throws. Non-identical and parallel slipknots with five throws demonstrated superior knot integrity compared with the same knot types with three throws. Safil had inferior knot properties as compared to the other materials, but Vicryl, Vicryl Plus and Novosyn behaved virtually the same. The type of knot and the use of different suture materials have important influence on the integrity of the knot. A high knot reliability is nowadays all the more important because of the frequent use of resorbable suture materials. The suture gradually loses strength during the resorption process, so that an extra margin of safety neutralizes the effect of this process. [ABSTRACT FROM AUTHOR]

This paper proposes an advanced lumber manufacturing model for real-time process control in saw mills in order to increase the yield of high value defect-free lumber. Detecting subsurface defects by scanning canted logs and generating a process plan to cut the logs can increase the yield of high-grade lumber in a saw mill industry. The defect detection process is performed using the ground-penetrating radar (GPR) system. More recently, a defect detection algorithm was developed to process GPR scanned data using the MATLAB® software. This research uses the distance and depth coordinates generated by the defect detection algorithm to develop the process plan that generates a cutting sequence for the resaw machine. The process plan is in the form of an algorithm written in MATLAB® with a simple user interface. The generated cutting sequence was validated by comparing to the conventional sawing sequence, where the operator of the resaw machine randomly performs the cutting of boards. An increase in the yield (in terms of dollar value) of about 27% was noticed using the GPR-based detection system which can map interior defects prior to sawing the log and enable an optimal sawing pattern. [ABSTRACT FROM AUTHOR]

A great deal of effort has been devoted to the inference of additive model in the last decade. Among existing procedures, the kernel type are too costly to implement for high dimensions or large sample sizes, while the spline type provide no asymptotic distribution or uniform convergence. We propose a one step backfitting estimator of the component function in an additive regression model, using spline estimators in the first stage followed by kernel/local linear estimators. Under weak conditions, the proposed estimator’s pointwise distribution is asymptotically equivalent to an univariate kernel/local linear estimator, hence the dimension is effectively reduced to one at any point. This dimension reduction holds uniformly over an interval under assumptions of normal errors. Monte Carlo evidence supports the asymptotic results for dimensions ranging from low to very high, and sample sizes ranging from moderate to large. The proposed confidence band is applied to the Boston housing data for linearity diagnosis. [ABSTRACT FROM AUTHOR]

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves. [ABSTRACT FROM AUTHOR]

The purpose of this study was to determine the biomechanical characteristics of 16 arthroscopic knots and to determine if locking knots have superior loop security compared to non-locking knots. Sixteen knot types were tied in arthroscopic fashion and tested on a materials testing system. Knots were cyclically loaded to 30 Newtons (N) for 20 cycles and then loaded to failure at 1.25 mm/s. Ten samples of each knot were tied using both #2 Ethibond and #1 PDS II. Load to ultimate failure, load to clinical failure, post-cyclic stiffness, cyclical elongation, ultimate displacement, loop security, and mode of failure were determined for each knot. Nicky’s Knot and the French Knot were most consistently ranked within the top five knot types for each of the biomechanical parameters. Locking knots did not improve loop security over non-locking knots. [ABSTRACT FROM AUTHOR]

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodiment” of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory. [ABSTRACT FROM AUTHOR]

The design and synthesis of figure--eight macrocycles are very scarce owing to the intricacies and lack of predictability from first principles. This review emphasizes on discrete macrocyclic systems both synthetic and natural with a defined figure eight knotted topology. In almost all the helical macrocycles, the helical arrangement is held by intramolecular hydrogen bonding or as a backbone requirement, but in all cases, a planar graph can be drawn and so these compounds are trivial from the topological stand point. Nature presents great deal of complexity in terms of structures in macromolecules like DNA and proteins and also displays intriguing topology in simple natural products. Patellamide, tawicyclamides, nosiheptide and thiostrepton are natural products with figure eight topology which shows interesting biological activity. Expanded porphyrins, Cu(II) complexes of thiomacrocycles, cyclic peptides and oligoesters are synthetic macrocycles showing intriguing topology. Analysis of structure and folding behaviour will enable chemists to design molecules with intriguing topology. [ABSTRACT FROM AUTHOR]

The article is about mathematical motifs of modern interest such as remarkable links and knots that were found at a 6th century Chennai temple. Introducing these mathematical objects we discuss their timeless and universal appeal as well as point to possible reasons for their symbolic use in ancient Indian temples. [ABSTRACT FROM AUTHOR]

Intravesical catheter knotting during micturating cystourethrography is a rare but recognized complication of the procedure. We were able to untangle a knot utilizing a fluoroscopically guided vascular guidewire. Following this success, a small study was performed using a model. Various types of guidewires and techniques were tested for different diameters of knots in order to predict the likelihood of success in this type of situation. [ABSTRACT FROM AUTHOR]

Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bézier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C 2 cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper. [ABSTRACT FROM AUTHOR]

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