New Stick Number Bounds from Random Sampling of Confined Polygons

The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings.Comment: 35 pages, 6 figure

Knot Types Used by Transformable and Rigid Linear Structural Systems

A knot is the joining place between two or more constructive elements in a framework or structure. They have a fundamental importance in the structure, according to their design they will be able to give a geometric configuration or another to the system and will also absorb certain forces or others. Depending on the movements they allow to the bars, there are rigid knots, articulated knots and slip knots. In this paper a study of cases about rigid knots or embedments used by structural systems so far will be presented. These types of knots prevent the rotation and movement of the constructive elements used for construction. In this paper also a study of cases about the articulated and slip knots used by transformable structural systems so far will be presented. An articulated knot allows the rotation but not the movement of the elements. A slip knot prevents movement in one of the three axes of the reference system, but not in the others, nor in the rotation between the elements. The research is focused in presenting a summary and comparison of rigid knots, articulated knots and slip knots that have been used in the structural design of some architecture. The union systems research will be crucial in this study. The investigation shows an important state of the art that provides technical solutions to apply on novel architectures based on rigid structural systems and articulated and slip structural systems. The research is useful to produce the current constructive solutions based on these constructive systems

Quipus and Witches’ Knots: The Role of the Knot in Primitive and Ancient Cultures

This essay in cultural anthropology provides a comprehensive view of the way primitive people in all parts of the world once utilized knots; mnemonic knots—to record dates, numbers, and cultural traditions; magic knots—to cure diseases, bewitch enemies, and control the forces of nature; and practical knots—to tie things and hold things together. In his discussion of mnemonic knots, the author analyzes the Peruvian quipus (or knot-calendars and knot-records) and suggests that the Inca astronomer-priests, known to have been accurate observers of the movements of the planets, may also have been able to predict the dates of lunar eclipses; and he shows how it is possible to manipulate the Ina abacus in accordance with the decimal system. His treatment of magic knots includes instances from Babylonian times to the present, with curious examples of the supernatural power attributed to the Hercules knot (i.e., the square knot) in Egypt, Greece, and Rome. His analysis of a little-known treatise on surgeons’ slings and nooses, written by the Green physician Heraklas, is the first detailed account of the specific practical knots used by the ancient Greeks and Romans. Quipus and Witches’ Knots, which is abundantly illustrated, often surprises the reader with the unexpected ways in which the once universal dependence of men on knots has left its mark on the language, customs, and thought of modern peoples. Description Cyrus Lawrence Day (1900–1968) was professor emeritus of English at the University of Delaware. His publications include The Songs of John Drydenand The Art of Knotting and Splicing. This Kansas Open Books title is funded by a grant from the National Endowment for the Humanities and the Andrew W. Mellon Foundation Humanities Open Book Program.https://digitalcommons.pittstate.edu/kansas_open_books/1015/thumbnail.jp

CelticGraph: Drawing Graphs as Celtic Knots and Links

Celtic knots are an ancient art form often attributed to Celtic cultures, used to decorate monuments and manuscripts, and to symbolise eternity and interconnectedness. This paper describes the framework CelticGraph to draw graphs as Celtic knots and links. The drawing process raises interesting combinatorial concepts in the theory of circuits in planar graphs. Further, CelticGraph uses a novel algorithm to represent edges as B\'ezier curves, aiming to show each link as a smooth curve with limited curvature.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Improving Knot Segmentation Using Deep Learning Techniques

In the context of Computed Tomography scanning of logs, accurate detection of knots is key for delivering a successful product. Reliable detection of knots in the sapwood is hard with traditional computer vision techniques, because of the different density conditions between sapwood and heartwood. The advancements provided by deep learning in the field of semantic image segmentation kick-started a new way of approaching such problems: deep neural networks can be trained on large amounts of labelled data and successfully employed in production environments to improve the performances on knot detection. Adapting state-of-the-art network architectures and using more than 10.000 labelled knots from pine and spruce logs, we were able to develop a new two-step approach for identifying knots in CT scans of logs with unprecedented accuracy while at the same time satisfying the time constraints that a real-time industrial application needs. The first step runs on the log’s axis, while the second runs on each candidate knot’s axis. False positives from the first step are very rare (even with dry/dried logs), so no computational power is wasted for non-existing knots. Using this approach, we are able to see the internal defects of a log in real time in the production chain without having to cut it first, therefore being able to optimize even more the output of the chain on each client’s requirements

Torus knots and Dunwoody manifolds

We obtain an explicit representation, as Dunwoody manifolds, of all cyclic branched coverings of torus knots of type $(p,mp\pm 1)$, with $p>1$ and $m>0$.Comment: 11 pages, 7 figures, to appear in the Siberian Mathematical Journa

RadixSpline: A Single-Pass Learned Index

Recent research has shown that learned models can outperform state-of-the-art index structures in size and lookup performance. While this is a very promising result, existing learned structures are often cumbersome to implement and are slow to build. In fact, most approaches that we are aware of require multiple training passes over the data. We introduce RadixSpline (RS), a learned index that can be built in a single pass over the data and is competitive with state-of-the-art learned index models, like RMI, in size and lookup performance. We evaluate RS using the SOSD benchmark and show that it achieves competitive results on all datasets, despite the fact that it only has two parameters.Comment: Third International Workshop on Exploiting Artificial Intelligence Techniques for Data Management (aiDM 2020

Maxwell's Theory of Solid Angle and the Construction of Knotted Fields

We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in $\mathbb{R}^3$. This is a purely geometric construction in which all of the properties of the entire knotted field derive from the geometry of the curve, and from projective and spherical geometry. We emphasise a fundamental homotopy formula as unifying different formulae for computing the solid angle. The solid angle induces a natural framing of the curve, which we show is related to its writhe and use to characterise the local structure in a neighborhood of the knot. Finally, we discuss computational implementation of the formulae derived, with C code provided, and give illustrations for how the solid angle may be used to give explicit constructions of knotted scroll waves in excitable media and knotted director fields around disclination lines in nematic liquid crystals.Comment: 20 pages, 9 figure