Volume Of Knots Research Articles

The tightest knot is not necessarily the smallest

In this paper, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its convex hull. We measure the convex hull volume of knots during the length annealing process, identifying local minima in the hull volume that arise due to buckling and symmetry breaking. We use [Formula: see text] torus knots as an illustrative example of a family of knots whose locally minimal-length embeddings are not necessarily ordered by volume. We identify several knots whose central curve has a convex hull volume that is not minimized in the ideal configuration, and find that [Formula: see text] has a non-ideal global minimum in its convex hull volume even when the thickness of its tube is taken into account.

Prediction models of knot volume inside the stem for Korean pine plantation

Based on 1207 knots from 49 sample trees of 29 standard plots of Korean pine plantations in Linkou and Dongjingcheng Forest Bureau of Heilongjiang Province, China, we extracted longitudinal sections of knots using the image processing software Digimizer and represented the shape of knots using two-dimensional scatter plots. According to the two-dimensional scatter plots, knots of Korean pine plantation were divided into three types: 1) alive knots (whole knot contained only sound knot portion); 2) non-occluded dead knots (whole knot contained both sound and loose knot portions); 3) occluded dead knots (the sound and loose portion of the knot were partially occluded by the bark). For all the three types of knots, the volume of sound knot was calculated by mathematical integral of the sound knot shape equation. The volume of loose knot was calculated using the volume equation of a cylinder. The total volume of knots was calculated as the sum of sound and loose knot volume. Finally, based on knot variables (diameter, relative height and total length of knots) and tree variable (diameter at breast height), we established the prediction models for sound knot volume, loose knot volume, and total volume of knot using the linear mixed model at plot level and tree level. Compared with fixed-effects model, the mixed effects models of the volume of sound knot, loose knot, and total knots provided more accurate parameter estimation, more uniform residual distribution, and higher model fitting precision. The validation results showed that prediction precision of each fixed-effect model was higher than 90%, while that of the mixed models with plot and tree effect was above 93%, indicating that the established model could well predict the volume of knot for Korean pine plantation.

Predicting knottiness of Scots pine stems for quality bucking

Stem shapes and wood properties are typically unknown at the time of harvesting. To date, approaches that integrate information about past tree growth into the harvesting and bucking process are rarely used. New models were developed and their potential demonstrated for stem bucking procedures for cut-to-length harvesters that integrate information about external and internal stem characteristics detected during harvesting. In total 221 stems were sampled from nine Scots pine (Pinus sylvestris L.) stands in Finland. The widths of rings 11−20 from the pith were measured using images taken from the end face of each butt log. The total volume of knots in each whorl was measured by using a 4D X-ray log scanner. In addition, 13 stems were test sawn, and the diameters of individual knots were measured from the sawn boards. A model system was developed for predicting the horizontal diameter of the thickest knot for each whorl along a stem. The first submodel predicts the knot volume profile from the stem base upwards, and the second submodel converts the predicted knot volume to maximum knot diameter. The results showed that the knottiness of stems of a given size may vary greatly depending on their early growth rate. The developed system will be used to guide logging operations to achieve more profitable bucking procedures.

In vitro evaluation of the knot-holding capacity and security, weight, and volume of forwarder knots tied with size-3 polyglactin 910 suture exposed to air, balanced electrolyte solution, or equine abdominal fat.

To evaluate the effect of exposure to a balanced electrolyte solution (BES), or equine abdominal fat on the knot-holding capacity (KHC), relative knot security (RKS), weight, and volume of forwarder knots versus surgeon's knots. 315 knots tied and tested in vitro. United States Pharmacopeia size-3 polyglactin 910 suture exposed to air (dry [control]), equine abdominal fat (fat-exposed), or BES (BES-exposed) was used to tie forwarder knots with 2, 3, and 4 throws and surgeon's knots with 5, 6, 7, and 8 throws. A universal materials testing machine was used to test the tensile strength of suture and knots to failure, and the KHC, RKS, weight, and volume of knots were determined. Forwarder knots had significantly higher KHC and RKS and lower volume, compared with surgeons' knots. Forwarder knots tied with fat-exposed suture had greater weight, but not volume, than did forwarder knots tied with dry or BES-exposed suture with the same number of throws. Results indicated that forwarder knots were superior to surgeon's knots when configured as start knots intended for continuous lines of suture. Exposure to media did not negatively affect mechanical or physical properties of forwarder knots and may improve specific biomechanical functions, including KHC and RKS.

Twist number and the alternating volume of knots

It was previously shown by the first author that every knot in [Formula: see text] is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot [Formula: see text] to be the minimum volume of any link [Formula: see text] in a natural class of alternating, hyperbolic links such that [Formula: see text] is ambient isotopic to a component of [Formula: see text]. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot.

Kelvin wave and knot dynamics on entangled vortices

In this paper, starting from Biot–Savart mechanics for entangled vortex-membranes, a new theory — knot physics — is developed to explore the underlying physics of quantum mechanics. Owning to the conservation conditions of the volume of knots on vortices in incompressible fluid, the shape of knots will never be changed and the corresponding Kelvin waves cannot evolve smoothly. Instead, the knot can only be split. The knot-pieces evolve following the equation of motion of Biot–Savart equation that becomes Schrödinger equation for probability waves of knots. The classical functions for Kelvin waves become wave-functions for knots. The effective theory of perturbative entangled vortex-membranes becomes a traditional model of relativistic quantum field theory — a massive Dirac model. As a result, this work would help researchers to understand the mystery in quantum mechanics.

Braids, complex volume and cluster algebras

We try to give a cluster algebraic interpretation of complex volume of knots. We construct the R-operator from the cluster mutations, and we show that it is regarded as a hyperbolic octahedron. The cluster variables are interpreted as edge parameters used by Zickert in computing complex volume.

The nature of the H2-emitting gas in the Crab nebula★

Understanding how molecules and dust might have formed within a rapidly expanding young supernova remnant is important because of the obvious application to vigorous supernova activity at very high redshift. In previous papers, we found that the H2 emission is often quite strong, correlates with optical low-ionization emission lines, and has a surprisingly high excitation temperature. Here we study Knot 51, a representative, bright example, for which we have available long slit optical and NIR spectra covering emission lines from ionized, neutral, and molecular gas, as well as HST visible and SOAR Telescope NIR narrow-band images. We present a series of CLOUDY simulations to probe the excitation mechanisms, formation processes and dust content in environments that can produce the observed H2 emission. We do not try for an exact match between model and observations given Knot 51's ambiguous geometry. Rather, we aim to explain how the bright H2 emission lines can be formed from within the volume of Knot 51 that also produces the observed optical emission from ionized and neutral gas. Our models that are powered only by the Crab's synchrotron radiation are ruled out because they cannot reproduce the strong, thermal H2 emission. The simulations that come closest to fitting the observations have the core of Knot 51 almost entirely atomic with the H2 emission coming from just a trace molecular component, and in which there is extra heating. In this unusual environment, H2 forms primarily by associative detachment rather than grain catalysis. In this picture, the 55 H2-emitting cores that we have previously catalogued in the Crab have a total mass of about 0.1 M_sun, which is about 5% of the total mass of the system of filaments. We also explore the effect of varying the dust abundance. We discuss possible future observations that could further elucidate the nature of these H2 knots.

A framework for modeling the dynamics of first-order branches and spatial distribution of knots in loblolly pine trees

A stochastic model to simulate the processes of initiation, diameter growth, death, and self-pruning of branches in loblolly pine ( Pinus taeda L.) trees is presented. Information on whorl formation and branch growth was obtained from destructive sampling of whorl sections from 34 trees growing under 10 different initial spacings. Three different components were modeled and hierarchically connected: whorl, branches, and knots. For each new growing season, whorls and branches are assigned stochastically along and around the stem. Thereafter, branch diameter growth is predicted as a function of relative location within the live crown and stem growth. The branch model was linked to an individual-tree growth model, PTAEDA3.1, to simulate the dynamics of first-order branches arising from the main stem. Information on (i) vertical trend of branch diameter along and around the stem, (ii) volume of knots (live and dead portions), and (iii) spatial location, size, and type (live and dead) of knots can be obtained. In its current stage, the framework allows evaluation of the quality of trees and sawlogs produced, inclusion of additional wood properties, and linkage with industrial conversion processes (e.g., sawing simulation). However, further research is needed to obtain data on branch dynamics to validate the overall performance of the model and improve developed submodels.

The hyperbolic volume of knots from quantum dilogarithm

The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particularexamples, it is argued that, for a hyperbolic knot (link), the absolute valueof this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate.

Factors relating to the volume of surgical knots

The suture's knot site represents the highest amount and density of foreign body material of a knotted suture loop. The volume of a knot is directly related to the total amount of inflammatory reaction surrounding the knot. We measured the volume of seven surgical knots frequently used in surgical practice. Two different suture materials and two different sizes were tested. Suture size was found to contribute significantly more to the volume of all knots tested than the number of throws of which the knots were composed. Knots made with coated multifilament polyglactin-910 were significantly larger than the knots made with monofilament nylon. Clinical implications of these findings are discussed. Surgical techniques keeping the volume of knots to a possible minimum, are encouraged.