Configurations Of Knots Research Articles

A statistical approach to knot confinement via persistent homology

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

Ideal geometry of periodic entanglements

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.

The Chula Knot: A New Sliding Locking Knot With a Special Property

There are many types of sliding locking knots used in arthroscopic surgery. Each type has its advantages and disadvantages. This technical note describes a new sliding locking knot, the Chula knot. This knot has been used for arthroscopic shoulder surgery in more than 500 cases at our hospital. The Chula knot has a special property in that it can be unfastened and retightened in case of premature locking and unintended loop loosening. This retensioning ability cannot be found in other configurations of arthroscopic knots. The Chula knot also has other benefits: It has high ultimate tensile strength and is easy to tie.

Arthroscopic sliding knot: How many additional half-hitches are really needed?

To evaluate the optimal number of additional half-hitches needed to achieve optimal knot-holding capacity (KHC) of lockable sliding knots. Mechanical testing study. Four configurations of arthroscopic knots (Duncan loop, Field knot, Giant knot, and SMC knot) were tested for their knot-holding capacity. For each knot configuration, 6 sequential knots were made including the initial sliding knot and an additional 5 knots by increasing the half-hitches 1 at a time. Each additional half-hitch was made as a reverse half-hitch with alternate posts. For each sequential knot configuration, 12 knots were made using No. 2 braided sutures. On the Servo-hydraulic materials testing system (Instron 8511; MTS, Minneapolis, MN), cyclic loading, load to clinical failure (3-mm displacement), load to ultimate failure, and mode of failure were measured. Most of the initial loops without additional half-hitches showed dynamic failure with cyclic loading. However, after 1 additional half-hitch, all 3 (SMC, Field, and Giant) knots showed resistance to dynamic cyclic load. After 2 additional half-hitches, the Duncan loop was secured without slippage from the cyclic loading test. The mean displacement after the end of cyclic loading decreased with each additional half-hitch. In particular, the SMC and Giant knot reached plateau at 0.1-mm or less displacement after 1 additional half-hitch. The Field knot and Duncan loop needed 3 additional half-hitches. The SMC knot and Duncan loop needed 1 additional half-hitch to reach greater than 80 N at clinical failure, whereas the other 2 knots needed 2 additional half-hitches. For the load exceeding 100 N for clinical failure, the SMC knot required 3 additional half-hitches and the other 3 knots needed 4 additional half-hitches. Addition of more than 3 half-hitches did not increase the load to clinical failure in the SMC knot. However, load to clinical failure increased up to 4 additional half-hitches in the other 3 knots (P < .05). The load to ultimate failure reached plateau when 3 or more additional half-hitches were made for all knot configurations. As the number of additional half-hitches increased, the mode of failure switched from pure loop failure (slippage) to material failure (breakage). The Duncan loop showed poor loop security--even with 5 additional half-hitches, some failed by slippage (17%). On the other hand, after 3 additional half-hitches, the 3 other knots showed greater than 75% of failure by material breakage mode (SMC and Field 92%, Giant 75%). Even with its own locking mechanism, a lockable sliding knot alone does not withstand the initial dynamic cyclic load. For all tested variables, the SMC knot required a minimum of 2 additional half-hitches. All knots showed a near plateau in knot security with 3 or more additional half-hitches. The Duncan loop may need more than 3 additional half-hitches for optimal security. The study shows that the knots tested needed at least 2 additional half-hitches, so this should become standard clinical practice.

Hyperbolic 2-fold branched coverings of links and their quotients

Many 3-manifolds can be represented as 2-fold branched coverings of links, but this representation is, in general, not unique. In the Seifert fibered case the problem is usually local: For example, if K is a Montesinos knot its 2-fold branched covering is Seifert fibered and there exists a complete system of local geometric modifications on K by which we can get every other Montesinos knot with the same 2-fold branched covering. On the other hand, if the 2-fold covering M of a knot is hyperbolic, the situation is globally determined by the structure of the isometry group of M. In this paper we develop a global approach for the case that M is hyperbolic and we study the orbifolds which are quotients of M by the action of a 2-group of isometries. This leads to a complete description of the geometry of the possible configurations of knots with the same 2-fold branched coverings. Moreover we are also able to settle the 2-component link case, which was still open, by finding an explicit bound on the number of in-equivalent 2-component links which have the same hyperbolic 2-fold branched coverings.

Significance of the internal locking mechanism for loop security enhancement in the arthroscopic knot

Purpose: Secure tissue fixation of an arthroscopic knot depends on the security of the initial loop and additional locking half-hitches. The purpose of this report is to evaluate the loop and knot security relative to the internal locking mechanism of arthroscopic slip knot. Type of Study: This is an experimental study designed for knot and loop security on a material testing system. The measurements were carried out in a blinded fashion in which the tester was not aware of the type of knot being tested. Methods: Five configurations of arthroscopic knots (overhand throw, Duncan loop, Revo knot, Tennessee slider, and SMC knot) were created around a 5-mm diameter metal bar using a No. 2 braided suture. For each knot configuration, 10 knots were tested for displacement at cyclic loading, load to clinical failure (3-mm displacement), ultimate failure load, and mode of failure on the servo-hydraulic material testing system (MTS 858 MiniBionix test system; NITS, Minneapolis, MN). A 1-way analysis of variance was used to determine the statistical difference in displacement at cyclic loading, load to clinical failure, and ultimate failure load between each knot configuration. Results: In the cyclic loading test, the average displacement of all knots was clinically insignificant, with the average being less than 0.24 mm. The overhand throw had the worst characteristic in the load to failure test. The other 4 knots had an optimal load to failure. The SMC knot, which has an internal locking mechanism, had good loop and knot security in the load to clinical failure and ultimate failure load (P < .05). The SMC and the Revo knots failed by knot breakage (material failure) whereas the other knots failed by knot slippage (loop failure). Conclusions: The internal locking mechanism flips the post strand to convert the loop strand into a new post strand while rerouting the original post strand around a new post. An arthroscopic knot with an internal locking mechanism can enhance loop security.

A topological invariant to predict the three-dimensional writhe of ideal configurations of knots and links.

We present herein a topological invariant of oriented alternating knots and links that predicts the three-dimensional (3D) writhe of the ideal geometrical configuration of the considered knot/link. The fact that we can correlate a geometrical property of a given configuration with a topological invariant supports the notion that the ideal configuration contains important information about knots and links. The importance of the concept of ideal configuration was already suggested by the good correlation between the 3D writhe of ideal knot configurations and the ensemble average of the 3D writhe of random configurations of the considered knots. The values of the new invariant are quantized: multiples of 4/7 for links with an odd number of components (including knots) and 2/7 plus multiples of 4/7 for links with an even number of components.

Geometry and Physics of Catenanes Applied to the Study of DNA Replication

The concept of ideal geometric configurations was recently applied to the classification and characterization of various knots. Different knots in their ideal form (i.e., the one requiring the shortest length of a constant-diameter tube to form a given knot) were shown to have an overall compactness proportional to the time-averaged compactness of thermally agitated knotted polymers forming corresponding knots. This was useful for predicting the relative speed of electrophoretic migration of different DNA knots. Here we characterize the ideal geometric configurations of catenanes (called links by mathematicians), i.e., closed curves in space that are topologically linked to each other. We demonstrate that the ideal configurations of different catenanes show interrelations very similar to those observed in the ideal configurations of knots. By analyzing literature data on electrophoretic separations of the torus-type of DNA catenanes with increasing complexity, we observed that their electrophoretic migration is roughly proportional to the overall compactness of ideal representations of the corresponding catenanes. This correlation does not apply, however, to electrophoretic migration of certain replication intermediates, believed up to now to represent the simplest torus-type catenanes. We propose, therefore, that freshly replicated circular DNA molecules, in addition to forming regular catenanes, may also form hemicatenanes.

Properties of ideal composite knots.

The shortest tube of constant diameter that can form a given knot represents the 'ideal' form of the knot. Ideal knots provide an irreducible representation of the knot, and they have some intriguing mathematical and physical features, including a direct correspondence with the time-averaged shapes of knotted DNA molecules in solution. Here we describe the properties of ideal forms of composite knots-knots obtained by the sequential tying of two or more independent knots (called factor knots) on the same string. We find that the writhe (related to the handedness of crossing points) of composite knots is the sum of that of the ideal forms of the factor knots. By comparing ideal composite knots with simulated configurations of knotted, thermally fluctuating DNA, we conclude that the additivity of writhe applies also to randomly distorted configurations of composite knots and their corresponding factor knots. We show that composite knots with several factor knots may possess distinct structural isomers that can be interconverted only by loosening the knot.

Physically-based stochastic simplification of mathematical knots

The article describes a tool for simplification and analysis of tangled configurations of mathematical knots. The proposed method addresses optimization issues common in energy based approaches to knot classification. In this class of methods, an initially tangled elastic rope is charged with an electrostatic like field which causes it to self repel, prompting it to evolve into a mechanically stable configuration. This configuration is believed to be characteristic for its knot type. We propose a physically based model to implicitly guard against isotopy violation during such evolution and suggest that a robust stochastic optimization procedure, simulated annealing, be used for the purpose of identifying the globally optimal solution. Because neither of these techniques depends on the properties of the energy function being optimized, our method is of general applicability, even though we applied it to a specific potential here. The method has successfully analyzed several complex tangles and is applicable to simplifying a large class of knots and links. Our work also shows that energy based techniques will not necessarily terminate in a unique configuration, thus we empirically refute a prior conjecture that one of the commonly used energy functions (J. Simon, 1994) is unimodal. Based on these results we also compare techniques that rely on geometric energy optimization to conventional algebraic methods with regards to their classification power.

An Algorithm for Constructing Configurations of Knots for Bivariate B-Splines

This paper is concerned with an algorithm for constructing suitable configurations of knot sets for bivariate B-splines on arbitrary planar domains. The objective is to determine positions for the knots realizing a stable basis of B-splines with optimal smoothness. Furthermore, we wish to reduce the number of B-splines not vanishing at any given point of the domain. Uniform knot configurations [W. Dahmen and C. A. Micchelli, Numerical algorithms for least squares approximation by multivariate B-splines, Numerical Methods of Approximation Theory, Birkhauser, Basel, 1981] are covered as a special case. The paper includes some numerical experiments that illustrate the performance of the algorithm.