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Journal articles on the topic 'Ideal knots'

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Published: 10 December 2022

Last updated: 29 January 2023

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1

GONZALEZ, O., and R. DE LA LLAVE. "EXISTENCE OF IDEAL KNOTS." Journal of Knot Theory and Its Ramifications 12, no. 01 (February 2003): 123–33. http://dx.doi.org/10.1142/s0218216503002354.

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Ideal knots are curves are that maximize the scale invariant ratio of thickness to length. Here we present a simple argument to establish the existence of ideal knots for each knot type and each isotopy class and show that they are C1,1 curves.

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2

Keener, J. P. "Knotted vortex filaments in an ideal fluid." Journal of Fluid Mechanics 211 (February 1990): 629–51. http://dx.doi.org/10.1017/s0022112090001732.

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Knotted closed-curve solutions of the equation of self-induced vortex motion are studied. It is shown that there are invariant torus knots which translate and rotate as rigid bodies. The general motion of ‘small-amplitude’ torus knots and iterated (cabled) torus knots is described and found to be almost periodic in time, and for some, but not all, initial data, the topology of the knot is shown to be invariant.

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3

CALLAHAN, PATRICK J., JOHN C. DEAN, and JEFFREY R. WEEKS. "THE SIMPLEST HYPERBOLIC KNOTS." Journal of Knot Theory and Its Ramifications 08, no. 03 (May 1999): 279–97. http://dx.doi.org/10.1142/s0218216599000195.

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While the crossing number is the standard notion of complexity for knots, the number of ideal tetrahedra required to construct the complement provides a natural alternative. We determine which hyperbolic manifolds with 6 or fewer ideal tetrahedra are knot complements, and explicitly describe the corresponding knots in the 3-sphere. Thus, these 72 knots are the simplest knots according to this notion of complexity. Many of these knots have the structure of twisted torus knots. The initial observation that led to the project was the abundance of knot complements with small Seifert-fibered Dehn fillings among the census manifolds. Since many of these knots have rather large crossing number they do not appear in the knot tables. Our methods, while ad hoc, yield some detailed information about the knot complements as well as the manifolds that arise from exceptional surgeries on these knots.

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4

Dobrowolski, Jan Cz, and Aleksander P. Mazurek. "Model Carbyne Knots vs Ideal Knots†." Journal of Chemical Information and Computer Sciences 43, no. 3 (May 2003): 861–69. http://dx.doi.org/10.1021/ci020063w.

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5

Tran, Anh T. "The strong AJ conjecture for cables of torus knots." Journal of Knot Theory and Its Ramifications 24, no. 14 (December 2015): 1550072. http://dx.doi.org/10.1142/s0218216515500728.

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The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cable knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cable knots over torus knots.

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6

Stott, Philip M., Lionel G. Ripley, and Michael A. Lavelle. "The Ultimate Aberdeen Knot." Annals of The Royal College of Surgeons of England 89, no. 7 (October 2007): 713–17. http://dx.doi.org/10.1308/003588407x205468.

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INTRODUCTION The Aberdeen knot has been shown to be stronger and more secure than a surgeon's knot for ending a suture line. No data exist as to the ideal configuration of the Aberdeen knot. The Royal College of Surgeons of England in their Basic Surgical Skills Course, 2002 recommended six throws. The aim of this experiment is to find the ideal combination of throws and turns. MATERIALS AND METHODS Aberdeen knots of various configurations were tied in O-PDS suture (Ethicon, Johnson and Johnson). Each configuration was tied 10 times. A materials testing machine was used to test the knots to destruction in a standardised manner. RESULTS The knots were seen to behave in two ways. They either slipped and unravelled, or broke. Knots tied with fewer than three throws were unreliable. Knots tied with three throws and two turns appear to be the strongest configuration. Adding further throws and turns does not increase the strength of an Aberdeen knot. CONCLUSIONS An Aberdeen knot tied with three throws and two turns is the ultimate Aberdeen knot.

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7

Schuricht, Friedemann, and Heiko von der Mosel. "Characterization of ideal knots." Calculus of Variations and Partial Differential Equations 19, no. 3 (April 1, 2004): 281–305. http://dx.doi.org/10.1007/s00526-003-0216-y.

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8

Katritch, Vsevolod, Wilma K. Olson, Piotr Pieranski, Jacques Dubochet, and Andrzej Stasiak. "Properties of ideal composite knots." Nature 388, no. 6638 (July 1997): 148–51. http://dx.doi.org/10.1038/40582.

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9

PIERAŃSKI, PIOTR. "IN SEARCH OF IDEAL KNOTS*." Computational Methods in Science and Technology 4, no. 1 (1998): 9–23. http://dx.doi.org/10.12921/cmst.1998.04.01.09-23.

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10

Rawdon, Eric J. "Can Computers Discover Ideal Knots?" Experimental Mathematics 12, no. 3 (January 2003): 287–302. http://dx.doi.org/10.1080/10586458.2003.10504499.

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11

DURUMERIC, OGUZ C. "LOCAL STRUCTURE OF IDEAL KNOTS, II CONSTANT CURVATURE CASE." Journal of Knot Theory and Its Ramifications 18, no. 11 (November 2009): 1525–37. http://dx.doi.org/10.1142/s0218216509007609.

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The thickness, NIR (K) of a knot or link K is defined to be the radius of the largest open solid tube one can put around the curve without any self intersections of the normal discs, which is also known as the normal injectivity radius of K. For C1,1 curves K, [Formula: see text], where κ(K) is the generalized curvature, and the double critical self distance DCSD (K) is the shortest length of the segments perpendicular to K at both end points. The knots and links in ideal shapes (or tight knots or links) belong to the minima of ropelength = length/thickness within a fixed isotopy class. In this article, we prove that NIR (K) = ½ DCSC (K), for every relative minimum K of ropelength in Rn for certain dimensions n, including n = 3.

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12

SIMON, JONATHAN K. "ENERGY FUNCTIONS FOR POLYGONAL KNOTS." Journal of Knot Theory and Its Ramifications 03, no. 03 (September 1994): 299–320. http://dx.doi.org/10.1142/s021821659400023x.

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We define a scale-invariant energy function for polygonal knots in ℜ3 based on the minimum distances between segments. The energy is bounded below by 2π. (minimum crossing number of the knot type). For each knot type, there exists an ideal number of segments, from which can be made an ideal conformation of the knot having minimum energy among all polygons realizing that knot type. Results leading to this include the following: The energy of an n-segment polygon is greater than n; if energy is bounded then ratios of edge lengths and angles are bounded away from zero; changing knot type requires passing an infinite energy barrier. We implement an algorithm, with the feature that preserving knot-type is guaranteed, to discover local energy-minimizing conformations for a given number of segments, and present some of these examples.

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13

Evans, Myfanwy E., Vanessa Robins, and Stephen T. Hyde. "Ideal geometry of periodic entanglements." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2181 (September 2015): 20150254. http://dx.doi.org/10.1098/rspa.2015.0254.

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Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo . This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the Shrink-On-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20–41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ -graphs and entangled tetrahedron- and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

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14

AITCHISON, IAIN R., and LAWRENCE D. REEVES. "ON ARCHIMEDEAN LINK COMPLEMENTS." Journal of Knot Theory and Its Ramifications 11, no. 06 (September 2002): 833–68. http://dx.doi.org/10.1142/s0218216502002001.

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We study a subclass of alternating links for which the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.

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15

CARLEN, M., and H. GERLACH. "FOURIER APPROXIMATION OF SYMMETRIC IDEAL KNOTS." Journal of Knot Theory and Its Ramifications 21, no. 05 (April 2012): 1250057. http://dx.doi.org/10.1142/s0218216511010115.

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Enforcing a specific symmetry group on a curve, knotted or not, is not trivial using standard interpolations such as polygons or splines. For a prescribed symmetry group we present a symmetrization process based on a Fourier description of a knot. The presence of symmetry groups implies a characteristic pattern in the Fourier coefficients. The relations between the coefficients are shown for five ideal knot shapes with their proposed symmetry groups.

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16

Olsen, Kasper W., and Jakob Bohr. "A principle for ideal torus knots." EPL (Europhysics Letters) 103, no. 3 (August 1, 2013): 30002. http://dx.doi.org/10.1209/0295-5075/103/30002.

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17

Boden, Hans U., Emily Dies, Anne Isabel Gaudreau, Adam Gerlings, Eric Harper, and Andrew J. Nicas. "Alexander invariants for virtual knots." Journal of Knot Theory and Its Ramifications 24, no. 03 (March 2015): 1550009. http://dx.doi.org/10.1142/s0218216515500091.

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Given a virtual knot K, we introduce a new group-valued invariant VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the zeroth ideal is a polynomial HK(s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial GK(s, t) introduced by Sawollek; Kauffman and Radford; and Silver and Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ϱ : VGK → GLn(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.

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18

Hyde, David A. B., Joshua Henrich, Eric J. Rawdon, and Kenneth C. Millett. "Knotting fingerprints resolve knot complexity and knotting pathways in ideal knots." Journal of Physics: Condensed Matter 27, no. 35 (August 20, 2015): 354112. http://dx.doi.org/10.1088/0953-8984/27/35/354112.

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19

Quaintenne, Gwenaël, Jan A. van Gils, Pierrick Bocher, Anne Dekinga, and Theunis Piersma. "Scaling up ideals to freedom: are densities of red knots across western Europe consistent with ideal free distribution?" Proceedings of the Royal Society B: Biological Sciences 278, no. 1719 (February 16, 2011): 2728–36. http://dx.doi.org/10.1098/rspb.2011.0026.

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Local studies have shown that the distribution of red knots Calidris canutus across intertidal mudflats is consistent with the predictions of an ideal distribution, but not a free distribution. Here, we scale up the study of feeding distributions to their entire wintering area in western Europe. Densities of red knots were compared among seven wintering sites in The Netherlands, UK and France, where the available mollusc food stocks were also measured and from where diets were known. We tested between three different distribution models that respectively assumed (i) a uniform distribution of red knots over all areas, (ii) a uniform distribution across all suitable habitat (based on threshold densities of harvestable mollusc prey), and (iii) an ideal and free distribution (IFD) across all suitable habitats. Red knots were not homogeneously distributed across the different European wintering areas, also not when considering suitable habitats only. Their distribution was best explained by the IFD model, suggesting that the birds are exposed to interference and have good knowledge about their resource landscape at the spatial scale of NW Europe, and that the costs of movement between estuaries, at least when averaged over a whole winter, are negligible.

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20

Maggioni, Francesca, and Renzo L. Ricca. "On the groundstate energy of tight knots." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2109 (June 24, 2009): 2761–83. http://dx.doi.org/10.1098/rspa.2008.0536.

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New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm. Results for torus knots are compared with previous work, and the groundstate energy spectrum of the first prime knots — up to 10 crossings — is presented and analysed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.

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21

Durumeric, Oguz C. "Local structure of ideal shapes of knots." Topology and its Applications 154, no. 17 (September 2007): 3070–89. http://dx.doi.org/10.1016/j.topol.2007.07.004.

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22

Pierański, P., and S. Przybył. "Quasi-quantization of writhe in ideal knots." European Physical Journal E 6, no. 2 (October 2001): 117–21. http://dx.doi.org/10.1007/s101890170011.

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23

CHO, JINSEOK, and JUN MURAKAMI. "THE COMPLEX VOLUMES OF TWIST KNOTS VIA COLORED JONES POLYNOMIALS." Journal of Knot Theory and Its Ramifications 19, no. 11 (November 2010): 1401–21. http://dx.doi.org/10.1142/s0218216510008443.

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For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume. We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume. Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

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24

Champanerkar, Abhijit, Ilya Kofman, and Timothy Mullen. "The 500 simplest hyperbolic knots." Journal of Knot Theory and Its Ramifications 23, no. 12 (October 2014): 1450055. http://dx.doi.org/10.1142/s0218216514500552.

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25

HILLMAN, J. A. "KNOT GROUPS AND SLICE CONDITIONS." Journal of Knot Theory and Its Ramifications 17, no. 12 (December 2008): 1511–17. http://dx.doi.org/10.1142/s0218216508006749.

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We introduce the notions of "k-connected-slice" and "π1-slice", interpolating between "homotopy ribbon" and "slice". Every high-dimensional knot group π is the group of an (n - 1)-connected-slice n-knot, for all n ≥ 3. However, if π is the group of an n-connected-slice n-knot, the augmentation ideal I(π) has deficiency 1 as a module, while (n + 1)-connected-slice n-knots are trivial. If π is the group of a π1-slice 2-knot and π' is finitely generated, then π' is free, and so def(π) = 1.

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26

RANKIN, STUART, ORTHO FLINT, and JOHN SCHERMANN. "ENUMERATING THE PRIME ALTERNATING KNOTS, PART I." Journal of Knot Theory and Its Ramifications 13, no. 01 (February 2004): 57–100. http://dx.doi.org/10.1142/s0218216504003044.

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The enumeration of prime knots has a long and storied history, beginning with the work of T. P. Kirkman [9,10], C. N. Little [14], and P. G. Tait [19] in the late 1800's, and continuing through to the present day, with significant progress and related results provided along the way by J. H. Conway [3], K. A. Perko [17, 18], M. B. Thistlethwaite [6, 8, 15, 16, 20], C. H. Dowker [6], J. Hoste [1, 8], J. Calvo [2], W. Menasco [15, 16], W. B. R. Lickorish [12, 13], J. Weeks [8] and many others. Additionally, there have been many efforts to establish bounds on the number of prime knots and links, as described in the works of O. Dasbach and S. Hougardy [4], D. J. A. Welsh [22], C. Ernst and D. W. Sumners [7], and C. Sundberg and M. Thistlethwaite [21] and others. In this paper, we provide a solution to part of the enumeration problem, in that we describe an efficient inductive scheme which uses a total of four operators to generate all prime alternating knots of a given minimal crossing size, and we prove that the procedure does in fact produce them all. The process proceeds in two steps, where in the first step, two of the four operators are applied to the prime alternating knots of minimal crossing size n to produce approximately 98% of the prime alternating knots of minimal crossing size n+1, while in the second step, the remaining two operators are applied to these newly constructed knots, thereby producing the remaining prime alternating knots of crossing size n+1. The process begins with the prime alternating knot of four crossings, the figure eight knot. In the sequel, we provide an actual implementation of our procedure, wherein we spend considerable effort to make the procedure efficient. One very important aspect of the implementation is a new way of encoding a knot. We are able to assign an integer array (called the master array) to a prime alternating knot in such a way that each regular projection, or plane configuration, of the knot can be constructed from the data in the array, and moreover, two knots are equivalent if and only if their master arrays are identical. A fringe benefit of this scheme is a candidate for the so-called ideal configuration of a prime alternating knot. We have used this generation scheme to enumerate the prime alternating knots up to and including those of 19 crossings. The knots up to and including 17 crossings produced by our generation scheme concurred with those found by M. Thistlethwaite, J. Hoste and J. Weeks (see [8]). The current implementation of the algorithms involved in the generation scheme allowed us to produce the 1,769,979 prime alternating knots of 17 crossings on a five node beowulf cluster in approximately 2.3 hours, while the time to produce the prime alternating knots up to and including those of 16 crossings totalled approximately 45 minutes. The prime alternating knots at 18 and 19 crossings were enumerated using the 48 node Compaq ES-40 beowulf cluster at the University of Western Ontario (we also received generous support from Compaq at the SC 99 conference). The cluster was shared with other users and so an accurate estimate of the running time is not available, but the generation of the 8,400,285 knots at 18 crossings was completed in 17 hours, and the generation of the 40,619,385 prime alternating knots at 19 crossings took approximately 72 hours. With the improvements that are described in the sequel, we anticipate that the knots at 19 crossings will be generated in not more than 10 hours on a current Pentium III personal computer equipped with 256 megabytes of main memory.

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27

Lin, Fanghua, and Yisong Yang. "Universal growth law for knot energy of Faddeev type in general dimensions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2098 (June 3, 2008): 2741–57. http://dx.doi.org/10.1098/rspa.2008.0128.

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The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of ‘ideal’ knots. In this paper, we show that the energy infimum E N stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration ( n ≥1) in general dimensions obeys the sharp fractional-exponent growth law , where the exponent p is universally rendered as , which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.

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28

Gonzalez, O., and J. H. Maddocks. "Global curvature, thickness, and the ideal shapes of knots." Proceedings of the National Academy of Sciences 96, no. 9 (April 27, 1999): 4769–73. http://dx.doi.org/10.1073/pnas.96.9.4769.

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29

Bartholomew, Paige, Shane McQuarrie, Jessica S. Purcell, and Kai Weser. "Volume and geometry of homogeneously adequate knots." Journal of Knot Theory and Its Ramifications 24, no. 08 (July 2015): 1550044. http://dx.doi.org/10.1142/s0218216515500443.

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We bound the hyperbolic volumes of a large class of knots and links, called homogeneously adequate knots and links, in terms of their diagrams. To do so, we use the decomposition of these links into ideal polyhedra, developed by Futer, Kalfagianni and Purcell. We identify essential product disks in these polyhedra.

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30

GELCA, RĂZVAN. "Non-commutative trigonometry and the A-polynomial of the trefoil knot." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 2 (September 2002): 311–23. http://dx.doi.org/10.1017/s0305004102006047.

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The non-commutative generalization of the A-polynomial of a knot of Cooper, Culler, Gillet, Long and Shalen [4] was introduced in [6]. This generalization consists of a finitely generated left ideal of polynomials in the quantum plane, the non- commutative A-ideal, and was defined based on Kauffman bracket skein modules, by deforming the ideal generated by the A-polynomial with respect to a parameter. The deformation was possible because of the relationship between the skein module with the variable t of the Kauffman bracket evaluated at −1 and the SL(2, C)-character variety of the fundamental group, which was explained in [2]. The purpose of the present paper is to compute the non-commutative A-ideal for the left- and right- handed trefoil knots. As will be seen below, this reduces to trigonometric operations in the non-commutative torus, the main device used being the product-to-sum formula for non-commutative cosines.

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31

SHIMA, AKIKO. "COLORINGS AND ALEXANDER POLYNOMIALS FOR RIBBON 2-KNOTS." Journal of Knot Theory and Its Ramifications 11, no. 03 (May 2002): 403–12. http://dx.doi.org/10.1142/s0218216502001706.

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32

Brasher, Reuben, Rob G. Scharein, and Mariel Vazquez. "New biologically motivated knot table." Biochemical Society Transactions 41, no. 2 (March 21, 2013): 606–11. http://dx.doi.org/10.1042/bst20120278.

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The knot nomenclature in common use, summarized in Rolfsen's knot table [Rolfsen (1990) Knots and Links, American Mathematical Society], was not originally designed to distinguish between mirror images. This ambiguity is particularly inconvenient when studying knotted biopolymers such as DNA and proteins, since their chirality is often significant. In the present article, we propose a biologically meaningful knot table where a representative of a chiral pair is chosen on the basis of its mean writhe. There is numerical evidence that the sign of the mean writhe is invariant for each knot in a chiral pair. We review numerical evidence where, for each knot type K, the mean writhe is taken over a large ensemble of randomly chosen realizations of K. It has also been proposed that a chiral pair can be distinguished by assessing the writhe of a minimal or ideal conformation of the knot. In all cases examined to date, the two methods produce the same results.

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33

LI, WEIPING. "KNOT AND LINK INVARIANTS AND MODULI SPACE OF PARABOLIC BUNDLES." Communications in Contemporary Mathematics 03, no. 04 (November 2001): 501–31. http://dx.doi.org/10.1142/s0219199701000470.

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In this paper, we show that the representation variety of the fundamental group of a 2n-punctured S2 with different conjugacy classes in SU(2) along punctures is a symplectic stratified variety from the group cohomology point of view. The representation variety can be identified with the moduli space of s-equivalence classes of stable parabolic bundles over the 2n-punctured S2 with corresponding weights along punctures, and also can be identified with the moduli space of gauge equivalence classes of SU(2)-flat connections with prescribed holonomies along punctures. We obtain an invariant of links (knots) from intersection theory on such a moduli space (a generalization of the signature of the link). We also study a SL2(C)-character variety of a knot K in S3 with fixed holonomy μ + μ-1 along the meridian of π1(S3\ K) (μ ∈ C*). The fixed-trace condition rules out the possibility of reducible representations with non-abelian image and the ideal point of irreducible representations via a generic perturbation. For knots without closed incompressible surfaces in S3 \ K, we show that there is a well-defined SL2(C)-knot invariant which is also related to them A-polynomial for special values μ ∈ U(1)\ {± 1}.

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34

Montesinos-Amilibia, José María. "On some groups related with Fox–Artin wild arcs." Journal of Knot Theory and Its Ramifications 25, no. 03 (March 2016): 1640007. http://dx.doi.org/10.1142/s0218216516400071.

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Two different topics are shown to be related. Some group presentations generalizing certain symmetric presentations found by Coxeter, and the ideal compactification of the sets obtained by lifting knots in 3-manifolds to their universal covering spaces.

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35

Ricca, Renzo L. "Topology bounds energy of knots and links." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2090 (November 5, 2007): 293–300. http://dx.doi.org/10.1098/rspa.2007.0174.

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In this paper, we determine two quantities, of geometric and topological character, that were left undetermined in two previous results obtained by Arnold (Arnold 1974 In Proc. Summer School in Diff. Eqs. at Dilizhan , pp. 229–256.) and Moffatt (Moffatt 1990 Nature 347 , 367–369) on lower bounds for the magnetic energy of knots and links in ideal fluids. For dissipative systems, a lower bound on magnetic helicity in terms of the average crossing number and a new relationship between rates of change of these two quantities are also determined.

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36

Hu, Kai, Baojin Wang, Yi Shen, Jieru Guan, and Yi Cai. "Defect identification method for poplar veneer based on progressive growing generated adversarial network and MASK R-CNN model." BioResources 15, no. 2 (March 16, 2020): 3041–52. http://dx.doi.org/10.15376/biores.15.2.3041-3052.

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As the main production unit of plywood, the surface defects of veneer seriously affect the quality and grade of plywood. Therefore, a new method for identifying wood defects based on progressive growing generative adversarial network (PGGAN) and the MASK R-CNN model is presented. Poplar veneer was mainly studied in this paper, and its dead knots, live knots, and insect holes were identified and classified. The PGGAN model was used to expand the dataset of wood defect images. A key ideal employed the transfer learning in the base of MASK R-CNN with a classifier layer. Lastly, the trained model was used to identify and classify the veneer defects compared with the back- propagation (BP) neural network, self-organizing map (SOM) neural network, and convolutional neural network (CNN). Experimental results showed that under the same conditions, the algorithm proposed in this paper based on PGGAN and MASK R-CNN and the model obtained through the transfer learning strategy accurately identified the defects of live knots, dead knots, and insect holes. The accuracy of identification was 99.05%, 97.05%, and 99.10%, respectively.

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Liu, Xin, and Renzo L. Ricca. "On the derivation of the HOMFLYPT polynomial invariant for fluid knots." Journal of Fluid Mechanics 773 (May 14, 2015): 34–48. http://dx.doi.org/10.1017/jfm.2015.231.

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By using and extending earlier results (Liu & Ricca,J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.

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Davidson, Carla, Przemyslaw Prusinkiewicz, and Patrick von Aderkas. "Description of a novel organ in the gametophyte of the fern Schizaea pusilla and its contribution to overall plant architecture." Botany 86, no. 10 (October 2008): 1217–23. http://dx.doi.org/10.1139/b08-085.

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Plant architecture is determined by cell division and growth, thus simulation models describing these processes are ideal for determining how local development produces the overall plant form. Because fern gametophytes are structurally simple, they are ideal for investigating the effects of cellular growth and division on plant form. In this work we examine the gametophytic development of Schizaea pusilla Pursh., a small, bog-adapted fern whose gametophyte forms as a mass of single-celled filaments. Using light and scanning electron microscopy we made detailed observations of gametophyte development to generate data for a simulation mechanical model of S. pusilla gametophyte development. To examine how plant architecture is an emergent property of cell division, we constructed a simulation model expressed using the formalism of L-systems. While developing a model of growth in this fern we discovered a previously undescribed structure that contributes to the architecture of this plant, which we term knots. We document the development of knots and demonstrate how they contribute to the overall plant architecture.

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Kurniawan, Abdy, Johny Malisan, and Muhammad Kadhafi Aznur. "Desain Kapal Landing Craft Utility Ideal untuk Trayek Perintis dan Tol Laut [Design Solution Ideal for Landing Craft Utility Ship for Pioneer Routes and Sea Toll]." Warta Penelitian Perhubungan 30, no. 2 (December 31, 2018): 133–43. http://dx.doi.org/10.25104/warlit.v30i2.831.

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AbstractThe condition of the open sea such as the area of Natuna becomes the challenge for the continuation of the service ship which causes the risky waves at certain times. Generally, it is known as North Monsoon. North Monsoon impacts on the cessation of the shipping activities for several months. In this condition, several small islands are in the isolated condition. This study aims to determine the ideal and adequate ship type based on the route serving for the isolated areas on the pioneer ship route and the sea toll route. The result in this research shows that the ship type which is suitable for the market share of 30% is landing craft utility type with 1000 DWT, service speed of 8 knots with the shipping frequency of 23 voyage per year.Keywords: North mansoon, landing craft utility, pioneer route, sea toll route, ship design. AbstrakKondisi perairan terbuka seperti wilayah Natuna, menjadi sebuah tantangan tersendiri terhadap kontinuitas pelayanan kapal yang disebabkan oleh kondisi gelombang yang rawan pada waktu tertentu. Umumnya hal ini dikenal dengan Angin Musim Utara. Angin Musim Utara berdampak pada berhentinya sebagian kegiatan pelayaran selama beberapa bulan. Dalam kondisi ini, beberapa pulau kecil berada dalam kondisi terisolasi. Kajian ini bertujuan untuk menentukan tipe kapal yang ideal dan sesuai dengan rute untuk melayani daerah terisolasi pada trayek kapal perintis dan trayek kapal tol laut. Hasil kajian menunjukkan bahwa tipe kapal yang sesuai dengan skenario market share 30% adalah tipe landing craft utility dengan ukuran 1000 DWT, kecepatan dinas 8 knot dengan frekuensi pelayaran sebanyak 23 voyage per-tahun.Kata kunci: Desain kapal, landing craft utility, trayek perintis, trayek tol laut, angin musim utara.

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Stubbs, Jack. "Hardwood Epicormic Branching—Small Knots but Large Losses." Southern Journal of Applied Forestry 10, no. 4 (November 1, 1986): 217–20. http://dx.doi.org/10.1093/sjaf/10.4.217.

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Abstract Epicormic sprouting was studied in a seed-tree regeneration cut and a selection cut, both made in mature, previously unmanaged, creek bottomland hardwoods in the lower coastal plain of South Carolina. The total exposure that seed trees received probably created near-ideal conditions for maximum epicormic sprouting, which allowed species to be ranked by sprouting propensity and degree of sprouting variation within species. Cherrybark oak produced the most sprouts, followed in descending order by swamp chestnut oak, Shumard oak, sweetgum, and yellow-poplar. White and green ash did not sprout, and Shumard oak and yellow-poplar were quite variable in sprouting incidence. The selection cut showed that in this stand, primarily composed of cherrybark oak and sweetgum, neither degree of release, crown class, nor direction of opening had an appreciable effect on sprout initiation or sprout numbers. Cherrybark oak sprouted less profusely than in the seed-tree cut, but sweetgum did not. Epicormic sprouting hazard is an important consideration in deciding whether to use some form of selection management. South. J. Appl. For. 10:217-220, Nov. 1986.

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Zhang, Bin, and Peter G. Wolynes. "Topology, structures, and energy landscapes of human chromosomes." Proceedings of the National Academy of Sciences 112, no. 19 (April 27, 2015): 6062–67. http://dx.doi.org/10.1073/pnas.1506257112.

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Chromosome conformation capture experiments provide a rich set of data concerning the spatial organization of the genome. We use these data along with a maximum entropy approach to derive a least-biased effective energy landscape for the chromosome. Simulations of the ensemble of chromosome conformations based on the resulting information theoretic landscape not only accurately reproduce experimental contact probabilities, but also provide a picture of chromosome dynamics and topology. The topology of the simulated chromosomes is probed by computing the distribution of their knot invariants. The simulated chromosome structures are largely free of knots. Topologically associating domains are shown to be crucial for establishing these knotless structures. The simulated chromosome conformations exhibit a tendency to form fibril-like structures like those observed via light microscopy. The topologically associating domains of the interphase chromosome exhibit multistability with varying liquid crystalline ordering that may allow discrete unfolding events and the landscape is locally funneled toward “ideal” chromosome structures that represent hierarchical fibrils of fibrils.

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Gils, Jan A. van, Bernard Spaans, Anne Dekinga, and Theunis Piersma. "FORAGING IN A TIDALLY STRUCTURED ENVIRONMENT BY RED KNOTS (CALIDRIS CANUTUS): IDEAL, BUT NOT FREE." Ecology 87, no. 5 (May 2006): 1189–202. http://dx.doi.org/10.1890/0012-9658(2006)87[1189:fiatse]2.0.co;2.

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43

Cerf, Corinne, and Andrzej Stasiak. "Linear relations between writhe and minimal crossing number in Conway families of ideal knots and links." New Journal of Physics 5 (July 7, 2003): 87. http://dx.doi.org/10.1088/1367-2630/5/1/387.

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Cerf, C., and A. Stasiak. "A topological invariant to predict the three-dimensional writhe of ideal configurations of knots and links." Proceedings of the National Academy of Sciences 97, no. 8 (April 11, 2000): 3795–98. http://dx.doi.org/10.1073/pnas.97.8.3795.

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45

Ziegler, Fabian, Nicole C. H. Lim, Soumit Sankar Mandal, Benjamin Pelz, Wei-Ping Ng, Michael Schlierf, Sophie E. Jackson, and Matthias Rief. "Knotting and unknotting of a protein in single molecule experiments." Proceedings of the National Academy of Sciences 113, no. 27 (June 23, 2016): 7533–38. http://dx.doi.org/10.1073/pnas.1600614113.

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Spontaneous folding of a polypeptide chain into a knotted structure remains one of the most puzzling and fascinating features of protein folding. The folding of knotted proteins is on the timescale of minutes and thus hard to reproduce with atomistic simulations that have been able to reproduce features of ultrafast folding in great detail. Furthermore, it is generally not possible to control the topology of the unfolded state. Single-molecule force spectroscopy is an ideal tool for overcoming this problem: by variation of pulling directions, we controlled the knotting topology of the unfolded state of the 52-knotted protein ubiquitin C-terminal hydrolase isoenzyme L1 (UCH-L1) and have therefore been able to quantify the influence of knotting on its folding rate. Here, we provide direct evidence that a threading event associated with formation of either a 31 or 52 knot, or a step closely associated with it, significantly slows down the folding of UCH-L1. The results of the optical tweezers experiments highlight the complex nature of the folding pathway, many additional intermediate structures being detected that cannot be resolved by intrinsic fluorescence. Mechanical stretching of knotted proteins is also of importance for understanding the possible implications of knots in proteins for cellular degradation. Compared with a simple 31 knot, we measure a significantly larger size for the 52 knot in the unfolded state that can be further tightened with higher forces. Our results highlight the potential difficulties in degrading a 52 knot compared with a 31 knot.

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Janušaitis, Rolandas, Valerijus Keras, and Jūratė Mockienė. "DEVELOPMENT OF METHODS FOR DESIGNING RATIONAL TRUSSES." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 9, no. 3 (September 30, 2003): 192–97. http://dx.doi.org/10.3846/13923730.2003.10531325.

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Striving for rationality and long-term reliability is seen in different periods of building activities. Application of linear programming methods has enabled to formalise this striving and to elaborate the necessary mathematical models. But later theoretical and practical investigations have disclosed that not always, when optimising in respect of one criterion, it is possible to obtain solutions rational in other aspects, and this stimulated the application of multicriteria optimization methods. It is useful in this case to apply the ideas of the game theory, game problems solving methods already applied in other building design fields. When adapting methods of the game theory to popular needs for truss designing, a criteria set involving 11 alternatives has been selected. Attempts have been made to find rational truss variants by applying different methods (method of proximity to an ideal point, Wald's and Hurwitz's methods). It has been found when using the method of proximity to an ideal point for rational truss designing that a truss with a sloping brace network and pivoted knots supported by a column and composed of rectangular box shapes is more valuable than other trusses. According to Wald's and Hurvitz's methods, among popular spans of 24 m such a truss is the truss with a lowered bottom chord.

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Taylor, H., and AW Grogono. "The constrictor knot is the best ligature." Annals of The Royal College of Surgeons of England 96, no. 2 (March 2014): 101–5. http://dx.doi.org/10.1308/003588414x13814021677638.

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Introduction An ideal ligature should tighten readily and remain tight. Ligature failure can be a critical complication of invasive procedures in human and veterinary surgical practice. Previous studies have tested various knots but not the constrictor knot. Methods A new test bench was employed to compare six ligatures using four suture materials. As tension in a ligature is not readily measured, the study employed a surrogate measurement: the force required to slide a ligature along a rod. Benchmark values tested each suture material wrapped around the rod to establish the ratio between this force and the ligature tension for each material. Each ligature was tested first during tightening and then again afterwards. The benchmark ratios were employed to calculate the tensions to evaluate which ligature and which suture material retained tension best. Results The model provided consistent linear relationships between the tension in the suture and the force required to pull the ligature along the rod. The constrictor knot retained tension in the ligature best (55–107% better than the next best ligature). Among the suture materials, polydioxanone had the greatest ability to retain the tension in a ligature and polyglactin the least. Conclusions The constrictor knot showed superior characteristics for use as a ligature, and should be introduced into teaching and clinical practice for human and veterinary surgery. The new test bench is recommended for future testing of ligatures as well as objective comparison of suture materials.

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Ahmad, Momin, Yi Luo, Christof Wöll, Manuel Tsotsalas, and Alexander Schug. "Design of Metal-Organic Framework Templated Materials Using High-Throughput Computational Screening." Molecules 25, no. 21 (October 22, 2020): 4875. http://dx.doi.org/10.3390/molecules25214875.

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The ability to crosslink Metal-Organic Frameworks (MOFs) has recently been discovered as a flexible approach towards synthesizing MOF-templated “ideal network polymers”. Crosslinking MOFs with rigid cross-linkers would allow the synthesis of crystalline Covalent-Organic Frameworks (COFs) of so far unprecedented flexibility in network topologies, far exceeding the conventional direct COF synthesis approach. However, to date only flexible cross-linkers were used in the MOF crosslinking approach, since a rigid cross-linker would require an ideal fit between the MOF structure and the cross-linker, which is experimentally extremely challenging, making in silico design mandatory. Here, we present an effective geometric method to find an ideal MOF cross-linker pair by employing a high-throughput screening approach. The algorithm considers distances, angles, and arbitrary rotations to optimally match the cross-linker inside the MOF structures. In a second, independent step, using Molecular Dynamics (MD) simulations we quantitatively confirmed all matches provided by the screening. Our approach thus provides a robust and powerful method to identify ideal MOF/Cross-linker combinations, which helped to identify several MOF-to-COF candidate structures by starting from suitable libraries. The algorithms presented here can be extended to other advanced network structures, such as mechanically interlocked materials or molecular weaving and knots.

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Semjon, Ján, Peter Demeč, and Jozef Svetlík. "Virtual Model of Tool Path for Milling Machine at Classical Design Base." Applied Mechanics and Materials 282 (January 2013): 235–41. http://dx.doi.org/10.4028/www.scientific.net/amm.282.235.

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This article focuses on issue of proposal ideal tool paths for machine tools. Model of machine consists from 6 basic knots where milling machine disposes spindle placed in the horizontal direction are. Based on mathematical analysis we can detect the movement of machine axes for uncertainty investigated. The calculated values can be compared with machine model developed in Computer - Aided Design. Defining the shape of workpiece as well as assigning an appropriate instrument can be determined by true value of precision workpiece. After substituting the values of specific dimensions we get the final position of vectors point for contact in tool coordinate systems at individual model solids.

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BALDWIN, JOHN A., and WILLIAM D. GILLAM. "COMPUTATIONS OF HEEGAARD-FLOER KNOT HOMOLOGY." Journal of Knot Theory and Its Ramifications 21, no. 08 (May 10, 2012): 1250075. http://dx.doi.org/10.1142/s0218216512500757.

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We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsváth and Sarkar. We review their construction, giving two examples that can be workout out by hand, and we explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, and we formulate a conjecture about the behavior of knot Floer homology under mutation, paying especially close attention to the Kinoshita–Terasaka knot and its Conway mutant. Finally, we discuss a conjecture of Rasmussen on relationship between Khovanov homology and knot Floer homology, and observe that it is consistent with our calculations.

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