Knots

The paper presents an ongoing collaborative project between a textile practitioner-researcher and a textile practitioner-mathematician that investigates the relationship between mathematical knot theory and knotted textiles. It examines how multiple monochrome textile knots may be characterised using mathematical analysis and how this in turn may facilitate the conceptualisation, design and production of knotted textiles.

Mathematical investigation of Nimkulrat’s knotted textile practice through the use of mathematical knot diagrams by Matthews revealed knot properties such as strand start/ end positions and strand active/passive roles which were indiscernible from the work alone. This approach led to a way of visualising knot designs using more than one colour prior to making. Further iterative design experimentation and material properties led to the creation of a new striped pattern and a three-dimensional artefact which will be exhibited.

The result of this current research phase illuminates the role of mathematics in making the knotting process explicit. It demonstrates the influence of mathematical analysis on craft practice and the significance of cross-disciplinary collaboration on the development of knotted pattern design.

Although surgery bears upon all sciences, yet, surgery itself can be seen as an art. What differentiates the best surgeon from the better one, is the ability of not just being able to manage the immediate and critical situations dynamically and to analyze the diseases interdisciplinarily , but his ability to perform perfect, aimful and economical coordination of operational movements. The refined technique of the handling and uniting the tissues -in the case of manual skills-is attainable by many practices, and the good surgeon works on the perfection of this technique in his daily operating activities. Dexterity and speed in tying knots correctly constitute an art which only practice can make perfect.

[Mini-catalog of exhibition:] Maedeup: Traditional Korean Ornamental Knots Exhibition & Demonstrations by Master Knot Maker Eunyoung Kim, Intangible Cultural Assets of Seoul City No. 13. Washington, D.C.: Korus House (and) U.S.-Korea Arts Foundation. [Text by Han Duk-soo, Gerald E. Connolly, Paul Michael Taylor, Kim Jun Il, and Jeong Won Seo.]  Catalog of the "Korus House" (Korean Cultural Center) exhibition of ornamental knots by Ms. Eun-young Kim, who was appointed Intangible
Cultural Assets of Seoul City No. 13. This exhibition and demonstration of ornamental knot-making illustrate the artistic achievements
and the craftsmanship of maedeup, traditional Korean ornamental knots. Created mostly from silk, each of the 33 basic types of maedeup has a different shape and each is named after an animal or object found in nature, such as a butterfly or chrysanthemum. Eun-young Kim began learning this craft in 1966 and studied under Hee-jin Kim. Maedeup were traditionally used, and are still used today, to decorate clothing, musical instruments and everyday objects. The maedeup, as an art form, has been designated Korea's "Important Intangible Cultural Property Number 22."

For the purposes of this article, climbing hitches will be assessed by three main criteria:
• how well the climbing hitch holds the climber in place for work positioning
By Mark Adams
• how easily the climbing hitch releases to allow the climber to descend and then grips once the climber has reached the next (usually lower) work station
• how easily the hitch breaks and advances when pushed by a slack tender or the climber’s hand and then grips once the climber has reached the next (usually higher) work station

The Situationist Times was a magazine edited and published by the Dutch artist Jacqueline de Jong during the years 1962–67. In its multilingual, transdisciplinary, and cross-cultural exuberance, it became one of the most exciting and playful magazines of the 1960s. Throughout its six remarkably diverse issues, The Situationist Times challenges the notion of what it means to be a situationist, as well as traditional understandings of culture in the broader sense and of how culture is created, formatted, and shared. These Are Situationist Times! provides an in-depth history of the magazine while probing its contemporary relevance. The book also presents the material De Jong assembled in the early 1970s in collaboration with Hans Brinkman for a never realized seventh issue of The Situationist Times, devoted to the game of pinball.
Lavishly illustrated and brimming with previously unseen archival material, These Are Situationist Times! presents new and compelling perspectives on situationism, experimental publishing, and artist-run magazines.

R. D. Laing’s critically neglected verse volume Knots (1970) is treated as a literary text and related to games, game theory and Cold War politics. The main focus is Laing’s use and view of language. He attempts, Zen-like, to reveal its conventionality and point towards another order of being. Knots participates in several genres and Laing - someone who sought to dissolve the doctor-patient distinction - transgresses what, he implies, are merely categories existing in language’s zone of illusion. His view of language relates, I argue, to his movement away from the speaking cure and towards greater interest in the body and pre-linguistic experience. While the countercultural Laing looks forward to the complete untangling of psychic, somatic and social knots, his presentation of such knots also suggests their unavoidability. If unavoidable, this text could help readers relate differently to their own knots and perhaps tie some more interesting ones.

REVIEW at: https://thebcreview.ca/2022/07/20/1528-laforet-croes-cordage/.

The “brother” to basketry, we have just published a 40-year update on ancient CORDAGE and KNOTS on the Northwest Coast of North America, with a focus on the 2000+ examples from the Ozette Village wet site. As one of the most common artifacts from Pacific Northwest wet (and shell midden) sites, cordage and knots play a key part in the dynamic Maritime Societies for over 10,000 years along the coast and riverways.

The flyer shows (in counterclockwise direction) 1. 3-strand, twisted cedar limb Ozette whale harpoon rope with a running noose knot on end, 2. Daryl Fedje, Director, holding a spruce and grass 3-strand braid string/thread dating to 10,700-years-old from a southern Haida Gwaii, B.C. wet site, 3. Makah Museum replica of the ancient Ozette house (in their museum) showing cedar limb 1-strand withes tying the house wall boards, slung between double pairs of poles, 4. 3-strand twisted cedar limb collar on a 700 year old Sunken Village wet site wooden wedge near Portland, OR, and 5. Ed Carriere, Suquamish Elder and Master Basket/Cordage Maker, shows how he stores his split cedar root in cordage loops for later use (these are common in ancient wet sites too). 

Please check it out at the publisher’s (JONA) web site: https://www.northwestanthropology.com/publications  and Amazon;

And see review in "The B.C. Review": https://thebcreview.ca/2022/07/20/1528-laforet-croes-cordage/

In this report ancient cordage artifacts from the Ozette Village archaeological site (45CA24) and other Northwest Coast water­logged sites are examined on three analytical levels. First, the cord­ age attribute (modes) are considered and compared between sites. Second, cordage stylistic/technological classes (types) and subclasses are created using a paradigmatic classification scheme. These cordage types and subtypes are compared between sites through the use of cluster analysis techniques. The results of these comparisons indicate a continuity of cordage styles in three separate regions of the Northwest Coast for the last 2000 to 3000 years, potentially indicating cultural continuity in these regions. These results were obtained when comparing basketry modes and types from these sites (Croes 1977b) and, in combination, these data support a hypothesis of regional cultural continuity in these regions. And third, a techno-functional classification of cordage artifacts sets from Ozette Village and other sites is considered. The Ozette cordage materials are especially useful for this purpose since they generally are recovered in their original context in the Ozette household. Comparisons of cordage functional sets from each Northwest Coast wet site demonstrate specific site-use differences. Not only can fishing stations be suggested by the abundance of nets or fishline leaders recovered at different sites, but the type of fishing practiced can be demonstrated. The three-level analysis of cordage from Ozette Village and other Northwest Coast wet sites demonstrates a special analytic value for cordage in Northwest Coast archaeological research.

“The sacred in architecture reveals itself beside a roadside stall, while you watch your batch of hot jalebis being made. The jalebi is a fabrication of the ubiquitous spiral, hence when you sink your teeth into one, with the juice exploding in your mouth; you know what spirals taste like…”

A month before my daughter Zahra was born, I was commissioned to write on the sacred in the architecture of all religions. I had turned down the commission arguing my expertise did not extend beyond Muslim Civilizations.

When Zahra was born, my father suggested we have jalebis to celebrate! (Jalebi is a dessert made by deep frying circular shapes of maida flour batter in a cauldron of boiling oil, and then soaked in sugar syrup). While biting into a hot jalebi, with its juice bursting in my mouth, in a flash my mind connected the movement of the chef’s hand while making jalebis with rituals of movement in the practice of several religions.

“Jalebi & The Knot of Sacred Architecture” was born at that instant; one of my quickest essays, it was written in a single draft (apologies for any unintended lapses and inadvertent faux pas).

In retrospect, growing up in multicultural India, the crossroad of world religions had sufficiently exposed me to multiple religious practices..

Published as "Contorurs of the Sacred" International Gallerie Vol.11 No.1 2008 pg.44-46
Final iteration forthcoming that restores its original title "Jalebi & The Knot of Sacred Architecture."

Cross Reference. See also:
"The Kaaba: Guarding The Centre, Generating The Circumference" in Articles Section
"Three Shores Of The Cube"  in Articles Section
"Medina Highway" in Poetry Section

Though scholars have amply documented the seminal role of Irenaeus of
Lyons in the development of Marian theology, Irenaeus uses a rare 
in Adversus Haereses 3.22.4 that has yet to be fully explored: “the knot of
Eve’s disobedience was untied through Mary’s obedience.” Why does Irenaeus choose such a seemingly weak image to describe the redemption of humanity? The present study shows through a close exegesis of the whole passage that Mary not only unties the knot of Satan, sin, and death, but she does so through an inverse tying of her own. Analysis of the ubiquitous literary, artistic, and material sources in the ancient Greco-Roman world reveals that the knot was a powerful and ambivalent symbol, connoting virginity and healing at the same time as bondage and constraint of the will. Irenaeus’s knot finally emerges as an eloquent and multivalent symbol for Mary’s crucial role in Christ’s recapitulation.

The article analyses a writing system of the Latvian language that has been scarcely researched before. The evidence of the usage of the knot script and song clews is present in the Latvian folklore, although it has been commonly considered as a metaphor. At the beginning of the 20th century there were individuals still using it prior to the conventional literacy increase due to the general education. Similar knot scripts can be found in several cultures around the globe. The article encompasses a study of previous researches and evidence in periodicals, Latvian community in Lithuania, education of commoners in Latvia, reciprocal comparison of the Latvian knot script writing systems, interview with a contemporary witness and experiment on reconstruction of the Latvian knot script.

Арзютов Д.В., Окотэтто Х.Х. Связывая вещи, животных и людей: к
социальной топологии ненецких узлов. // Археология Арктики. Вып.5./ Отв. ред. Н.В.Фёдорова. Салехард, 2018. С. 89-106.

Arzyutov, Dmitry V., and Khadri Okotetto. 2018. “Svi͡azyvai͡a Veshchi, Zhivotnykh i Li͡udeĭ: K Sot͡sial’noĭ Topologii Nenet͡skikh Uzlov [Tying Things, Animals and Humans Together: Towards Social Topology of Nenets Knots].” In Arkheologii͡a Arktiki, edited by Natalia V. Fedorova, 5:89–106. Salekhard.

In this paper we study the chests with the relics of San Vitores. The inner urn is Gothic, was made, around 1465, probably in Bruges (Flanders), and is adorned with wrought iron and leather worked with the cuir-ciselé technique. The exterior ark was made in Burgos in 1525. It presents Velasco family coats of arms and ornaments of knots, crosses and anagrams of Christ and Mary.
Estudiamos las arquetas con las reliquias de San Vitores. La urna interior es gótica, fue realizada, en torno a 1465, probablemente en Brujas (Flandes), y está adornada con hierro forjado y cuero trabajado con la técnica del cuir-ciselé. El arca exterior se hizo en Burgos en 1525. Presenta escudos de la familia Velasco y ornato de lazos, cruces y anagramas de Cristo y María.

Data derived from empirical studies, coupled with modeling and simulation techniques, were used to compare tree and product quality from two stands of small-diameter ponderosa pine trees growing in northern California: one plantation, the other natural. The plantation had no management following establishment, and the natural stand had no active management. Fifty trees of similar diameter classes were selected from each site, measured, bucked into logs, and sawn into boards, and the boards were scanned for defects. Trees from the plantation stand demonstrated less variability in tree height, crown length, and age structure. The plantation trees were on average 4 years younger than their natural cohorts, yet for all but the smallest diameter there were no significant differences between mean tree height or crown length. Predicted merchantable volumes differed significantly for the largest diameter class. Merchantable volume was estimated to be 5% greater for the plantation trees than...

Comment la musique crée-t-elle l'espace ? Quels types de structures spatiales la musique construit-elle ? Ces questions sont liées à ce que j'appelle la « visualisation de la musique dans l'espace », mais il nous faut d'abord nous demander : « quelle est le τέλος de la musique ? » Autrement dit, quel est le sens de la musique ? Comment la musique peut-elle être la musique ? La musique est-elle encore la musique lorsque nous ne l'écoutons pas ? A cet égard, interroger la musique invite à nous interroger nous-mêmes : comment comprenons-nous la musique ? La compréhension et la perception de la musique supposent celles des relations entre durées et instants, c'est-à-dire les relations du temps. La musique, l'espace, le temps, l'instant, la durée, la perception et la compréhension seront ainsi les concepts cruciaux de notre discussion. Le coeur de la discussion sur la musique, ou plutôt de la compréhension de la musique, est situé dans le concept de distinction. De quels types de distinction parlons-nous et pourquoi la distinction est-elle importante pour notre discussion ? La distinction est le début et le fondement de la compréhension, parce que la distinction révèle des frontières. A chaque moment du présent, la formation des frontières est créatrice d'espace(s) hors du vide. Cet événement du présent permet notre compréhension de la temporalité. Nous verrons que de tels espaces sont imaginaires, de même que cette temporalité. La distinction se trouve donc au coeur de notre discussion parce que la musique réalise dans notre temps et espace ordinaires la temporalité et la spatialité idéalisées. Par la musique, l'espace imaginaire remplit l'espace ordinaire. La musique est ainsi la réalisation de temporalités et d'espaces imaginaires, impliquant également les mathématiques et la conscience.

This paper attempts to explore a possibility to visualize the structure of time-consciousness in a knot shape. By applying Louis Kauffman's knot-logic, the consistency of subjective consciousness, the plurality of now's, and the necessary relationship between subjective and intersubjective consciousness will be represented in topological space. For a long time, I believed that time must flow like this, without much doubt: Figure 1 But, does it? What if time's arrow doesn't flow straight forward towards the future? I couldn't be sure if time doesn't flow like this, for example, in the shape of a knot:

A mathematician and a textile practitioner both with an interest in textile knotting and a focus on research in textile design have collaborated to analyze textile practice and realize novel knot designs through mathematical characterization. The design process has made use of century-old drawing techniques used to represent knots and braids. This paper demonstrates how the manipulation of these diagrams was utilized as a design tool to generate and visualize novel patterns.

Power and meaning of the "knot of Heracles". Potere e significato del "Nodo di Ercole", un incredibile amuleto che ha attraversato indenne il tempo. Indicatore-guida per la comprensione di iconografie antiche, soprattutto quando giunte a noi in maniera parziale.

We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or (2, m)torus knot or link substrate. We show that all knotted or linked products fall into a single family, and prove that the size of this family grows linearly with the cube of the minimum number of crossings. Additionally, we prove that the only possible products of an unknot substrate are either clasp knots and links or (2, m)-torus knots and links. Finally, in the (common) case of (2, m)-torus knot or link substrates whose products have minimal crossing number m + 1, we prove that the types of products are tightly prescribed, and use this to examine previously uncharacterized experimental data.

A fundamental problem in computational biology which has been widely studied in the last decades is the reconstruction of evolutionary trees from biological data. Unfortunately, almost all its known formulations are NP-hard. The compelling need for having efficient computational tools to solve this biologcal problem has brought a lot of attention to the analysis of the quartet paradigm for inferring evolutionary trees [1, 2, 8]. Given a quartet of taxa {a, b, c, d}, there are 3 possible degree-3 trees connecting the taxa as terminals. Each such tree is called a quartet topology. The quartet methods proceed by first estimating the topology of each quartet of taxa and then recombining the inferred quartet topologies into an evolutionary tree. A major difficulty in this approach derives from the fact that quartet topology inference methods often make mistakes, and thus may result in a set Q of quartet topologies that is inconsistent with any evolutionary tree. Therefore, the problem of...

Application of nonparametric and semiparametric regression techniques to high-dimensional time series data has been hampered due to the lack of effective tools to address the "curse of dimensionality." Under rather weak conditions, we propose spline-backfitted kernel estimators of the component functions for the nonlinear additive time series data that are both computationally expedient so they are usable for analyzing very high-dimensional time series, and theoretically reliable so inference can be made on the component functions with confidence. Simulation experiments have provided strong evidence that corroborates the asymptotic theory.

This essay explores the Mathematics of Charles Sanders Peirce. We concentrate on his notational approaches to basic logic and his general ideas about Sign, Symbol and diagrammatic thought. In the course of this paper we discuss two notations of Peirce, one of Nicod and one of Spencer-Brown. Needless to say, a notation connotes an entire language and these contexts are elaborated herein. The first Peirce notation is the portmanteau (see below) Sign of illation. The second Peirce notation is the form of implication in the existential graphs (see below). The Nicod notation is a portmanteau of the Sheffer stroke and an (overbar) negation sign. The Spencer-Brown notation is in line with the Peirce Sign of illation. It remained for Spencer-Brown (some fifty years after Peirce and Nicod) to see the relevance of an arithmetic of forms underlying his notation and thus putting the final touch on a development that, from a broad perspective, looks like the world mind doing its best to remember the significant patterns that join logic, speech and mathematics. The movement downward to the Form ("we take the form of distinction for the form."[9, Chapter 1, page 1]) through the joining together of words into archetypal portmanteau Signs can be no accident in this process of return to the beginning. We study a system of logic devised by Peirce based on a single Sign for inference that he calls his 'Sign of illation". We then turn to Peirce's Existential Graphs. The Existential Graphs lead to a remarkable connection between the very first steps in Logic and mirror plane symmetries of a "Logical Garnet" [30] in three dimensional space. Peirce's ideas about these graphs are related to his ideas about infinity and infinitesimals, and with his more general philosophy that regards a human being as a Sign. It is the intent of this paper to bring forth these themes in both their generality and their particularity. It is amazing that three dimensional geometry is closely allied to the first few distinctions of Logic. It is my intent in this paper to make that aspect crystal clear.

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and knot primeness detection.

Se evaluaron los medios de cultivo Hussey-Stacey (A), Hendrix et al. (H), Atkinson et al. (AT) y mitad de Murashyge y Skoog (½MS) para la propagación in vitro de semillas y explantes de Solanum mammosum, S. marginatum, S. hirtum y S. umbellatum. En la fase de propagación sexual se evaluó el porcentaje de germinación, número de raíces, longitud de planta, días a formación de hojas y raíces, días a morfogénesis completa y materia seca. Se determinó el tipo de morfogénesis a través de callos, vástago y plantas totalmente formadas. Se estudiaron el número y longitud de brotes, número de raíces y hojas, producción de materia seca y días a morfogénesis completa. Para S. mammosum el mejor medio de germinación fue ½MS, para el desarrollo de plantas fue A y para propagación vegetativa, A-Nudos (Medio A con explantes tipo nudos). En el mismo orden, para S. marginatum fueron los medios A, H y ½MS y/o H con nudos, mientras que en S. hirtum fueron H,A y H-nudos. Para S. umbellatum no se encontraron diferencias en germinación entre H, ½MS y A. Para la formación de plantas el mejor medio fue ½ MS y para propagación vegetativa fue H-Nudo.

Se evaluaron los medios de cultivo Hussey-Stacey (A), Hendrix et al. (H), Atkinson et al. (AT) y mitad de Murashyge y Skoog (½MS) para la propagación in vitro de semillas y explantes de Solanum mammosum, S. marginatum, S. hirtum y S. umbellatum. En la fase de propagación sexual se evaluó el porcentaje de germinación, número de raíces, longitud de planta, días a formación de hojas y raíces, días a morfogénesis completa y materia seca. Se determinó el tipo de morfogénesis a través de callos, vástago y plantas totalmente formadas. Se estudiaron el número y longitud de brotes, número de raíces y hojas, producción de materia seca y días a morfogénesis completa. Para S. mammosum el mejor medio de germinación fue ½MS, para el desarrollo de plantas fue A y para propagación vegetativa, A-Nudos (Medio A con explantes tipo nudos). En el mismo orden, para S. marginatum fueron los medios A, H y ½MS y/o H con nudos, mientras que en S. hirtum fueron H,A y H-nudos. Para S. umbellatum no se encontraron diferencias en germinación entre H, ½MS y A. Para la formación de plantas el mejor medio fue ½ MS y para propagación vegetativa fue H-Nudo.

The article analyses a writing system of the Latvian language that has been scarcely researched before. The evidence of the usage of the knot script and song clews is present in the Latvian folklore, although it has been commonly considered as a metaphor. At the beginning of the 20th century there were individuals still using it prior to the conventional literacy increase due to the general education. Similar knot scripts can be found in several cultures around the globe. The article encompasses a study of previous researches and evidence in periodicals, Latvian community in Lithuania, education of commoners in Latvia, reciprocal comparison of the Latvian knot script writing systems, interview with a contemporary witness and experiment on reconstruction of the Latvian knot script. // Keywords: knot script, knot records, quipu, Latvia, Latvian)

Given at: 'The Prehistory of Curiosity' Workshop, IUB, April 7-8.

This discussion paper used the figure of the knot (often described as curious in Middle English texts) to think about archival and conceptual ways of thinking through curiosity in the late Middle Ages.

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobás-Riordan-Tutte polynomial generalizes the Tutte polynomial of graphs to graphs that are embedded in closed oriented surfaces of higher genus.

The tension knot (or tensionless knoth), used precisely to pull zip lines, was proposed a few years ago and was consolidated in the industry thanks to its excellent performance in some specific contexts. However, like everyone else, it is important to know its scope and limitations, so that these announced advantages do not become problems and risks during the maneuver. The most important aspects to consider are the conditions of use, together with the limitations determined by the physical behavior of the rope with which the knot is made. Of course, only depending on the interaction of all these factors could we determine advantages, disadvantages and risks.

The probability that a random walk or polygon in the 3-space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, called the ephemeral knot. As is the case with knots, we prove that any topological knot type occurs as the ephemeral knotted portion of a slipknot. 1

This paper is about vislialization of illlisic 111 iinaginary space. This visualization is rather a mathematical-geometrical visualization which reveals the structure of our time understanding. The aim of this discussioil is to understand inusic, or inore precisely, to understilild how we understand inusic. Again this understaildiilg doesn't refer to an ilitellectual understanding, which is like 'understanding u h a t this implies.' I use tliis term 'comprehension' here in the episteniological sense, therefore. meaning. more or less. ((perception.)) Namely. it is t o investigate how we listell t o music knowing that it's rnusic. I11 tliis paper, I will discuss what is music. and in doing so I talk about 'distinction' and 'now' and the triadic structure of the original unity which leads us hack t o the discussio~l of t h e structure of the consciousness of music. T l~e n I will present possibilities to visualize the music. or. more precisely. the liilderstanding of music in the forin of a tetrahedroll and how the form of a tetrahedron can be transforlrled into the forin of a knot. Here, I'd like to start with the following question : what is the telos of niusic'? I11 other words. what is the teleology. the sense of inusic :' Or, very simply stated. hou call 111lisic be music ' ? Carl music be inlisic even when we are not listening 7 I11 this regard, im-estigatiiig music is in\-estigating our self : how d o we understand music ? hIusic can be comprehended as illusic only when inusic is heard as music by the subject The subject who understands music as music lies in the core of the ontological status of music. The epistemology of music by a conscious subject point a t the fact that the discussion of inusic lies within the realm of the discussion of subjectivity. However, the subjective uiiderstanding of music is different than my subjective opinion on a ccrtain issue opposed t o an objective viewpoint. This is coinparable t o our perception of colors. Colors are not absolute properties. Perceptioil of color ih subjective in the sense that the colors in the nature are the reflections of the different spectra of the sun rays that are dependent on our physical sensory