Knotted Strings Research Articles

Efficiency of non-identical double helix patterns in minimizing ropelength of torus knots

The ropelength of a knotted string with volume is defined as the ratio of the length of its central curve to the radius of its sectional disc. In a physical context, achieving minimal ropelength corresponds to a state of minimal potential energy, and geometrically, it signifies a tightly-packed conformation. The quest to establish a connection between the topological complexity of knotted strings and their minimal ropelength has persisted into recent years. In this paper, we introduce a new upper bound on the minimal ropelength of (2, n)-torus knots and links: Rop(T(2, n)) ≤ 7.3163 Cr(T(2, n)) + 17.1657. This upper bound is derived from a torus knot conformation constructed based on a tightened pattern of double helix with non-identical radii of winding. A comparative analysis with conformations generated from a superhelix and a circular helix underscores the efficiency of the non-identical double helix pattern, particularly when it appears as a long repeated motif in knotted strings.

Book Review: The Inka Empire: A Multidisciplinary Approach edited by Izumi Shimada

Izumi Shimada, ed. The Inka Empire: A Multidisciplinary Approach . Austin: University of Texas Press, 2015. Pp. 392. $75.00; ISBN: 978-0-292-76079-0. The Inka Empire or Tawantinsuyu (the Quechua name for the “realm of four parts”) was the largest political system to develop in the New World. Tawantinsuyu extended over 2,000,000 square kilometers and encompassed at least 86 ethnic groups from Argentina and Chile to the current border between Ecuador and Colombia. Intriguingly, the Inka ruled their empire without wheeled technologies, markets, money, or phonetic writing systems. Instead, they kept the records of the empire in a complex system of knotted strings called khipus . For that reason, much of what we know of this interesting polity comes from post-conquest sources. This breathtaking collaborative volume, edited by Izumi Shimada, seeks to offer a holistic vision of the Inka empire, placing different disciplines, methods, sources, and perspectives in dialogue. It includes 19 chapters by 23 authors that consider linguistic and genetic evidence along with material culture and historical documents to study Inka origins, imperial infrastructure, administrative strategies, agricultural technology, accounting, architecture and landscape intervention, and political organization, among other aspects. The volume is divided into five parts, prefaced by a useful introductory chapter by Shimada that lays out the book’s aims and …

Tackling tangledness of cosmic strings by knot polynomial topological invariants

Cosmic strings in the early universe have received revived interest in recent years. In this paper, we derive these structures as topological defects from singular distributions of the quintessence field of dark energy. Our emphasis is placed on the topological charge of tangled cosmic strings, which originates from the Hopf mapping and is a Chern–Simons action possessing strong inherent tie to knot topology. It is shown that the Kauffman bracket knot polynomial can be constructed in terms of this charge for unoriented knotted strings, serving as a topological invariant much stronger than the traditional Gauss linking numbers in characterizing string topology. Especially, we introduce a mathematical approach of breaking-reconnection which provides a promising candidate for studying physical reconnection processes within the complexity-reducing cascades of tangled cosmic strings.

Introduction: Andeans Articulating Colonial Worlds

Whether they be the painted walls and highly ‘dressed’ altars of Andean churches, or the collective knotted strings and litigation proceedings housed in locked chests in local Indian municipal councils, artifacts conjure up cultural interstices instrumental for discerning the “fine grain” of Spanish colonialism in the Andes. After the early years of the Spanish presence, the Andes became a mosaic of multifarious articulations of indigenous, European, and African ways of thinking, living, and weaving a social order that continued to change over time and across the geographical space of the viceroyalty of Peru.

OS2003 デジタル画像相関法による細線結び目の変形計測

It is well known that the tensile strength of knotted strings is weaker than the material strength of the strings. However, the strength of the knotted strings has not fully understood from the scientific viewpoint. In the present study, digital image correlation method is applied to examine the strength and characteristics of knots, especially overhand knot and figure-of-eight knot. The result shows that the shearing strain at the shoulder part of the overhand knot is larger than that of the figure-of-eight knot. This result is in good agreement with the result of tensile tests, which shows that the fracture stress of the overhand knot is lower than that of the figure-of-eight knot.

Crocodile Hunt

Crocodile Hunt

Education as Re-Embedding: Stroud Communiversity, Walking the Land and the Enduring Spell of the Sensuous

How we know, is at least as important as what we know: Before educationalists can begin to teach sustainability, we need to explore our own views of the world and how these are formed. The paper explores the ontological assumptions that underpin, usually implicitly, the pedagogical relationship and opens up the question of how people know each other and the world they share. Using understandings based in a phenomenological approach and guided by social constructionism, it suggests that the most appropriate pedagogical method for teaching sustainability is one based on situated learning and reflexive practice. To support its ontological questioning, the paper highlights two alternative culture’s ways of understanding and recording the world: Those of the Inca who inhabited pre-Columbian Peru, which was based on the quipu system of knotted strings, and the complex social and religious system of the songlines of the original people of Australia. As an indication of the sorts of teaching experiences that an emancipatory and relational pedagogy might give rise to, the paper offers examples of two community learning experiences in the exemplar sustainable community of Stroud, Gloucestershire in the United Kingdom where the authors live.

Spin-charge separation, conformal covariance and the [formula omitted] Yang–Mills theory

In the low energy domain of four-dimensional SU ( 2 ) Yang–Mills theory the spin and the charge of the gauge field can become separated from each other. The ensuing field variables describe the interacting dynamics between a version of the O ( 3 ) nonlinear σ-model and a nonlinear Grassmannian σ-model, both of which may support closed knotted strings as stable solitons. Lorentz transformations act projectively in the O ( 3 ) model which breaks global internal rotation symmetry and removes massless Goldstone bosons from the particle spectrum. The entire Yang–Mills Lagrangian can be recast into a generally covariant form with a conformally flat metric tensor. The result contains the Einstein–Hilbert Lagrangian together with a nonvanishing cosmological constant, and insinuates the presence of a novel dimensionfull parameter in the Yang–Mills theory.

Knot-Words or Not Words

Knot-Words or Not Words Frank Salomon, The Cord Keepers: Khipus and Cultural Life in a Peruvian Village. Duke University Press, 2004, 331 pp. The Khipu Enigma I felt a pang the other day when I read about multi-spectral imaging techniques that are recovering inscriptions on papyrus fragments over two thousand years old. Rescued from an ancient trash heap south of Cairo, this collection of 400,000 bits of trash has sat in Oxford for over a century, painstakingly curated, and yet the vast majority burned and discolored beyond legibility. With the new technique, a by-product of satellite technology, words from the distant past are coming into focus-seventy lines of a lost play by Sophocles, thirty lines from a seventh century B.C. poet named Antilochos. We can look forward to new (to us) poems by Pindar, Sappho and Euripides, writings by early Christians-and of course the detritus of everyday Greek-Egyptian life, like tax returns and horoscopes. So why should this interesting news send a pang through my heart? Because I study the Andes, home to a great civilization whose trash heaps and tombs yield up no ancient words. As far as words are concerned, we have to make do with Spanish accounts of post-Conquest lnka society, and with the very rare colonial era documents written by native Andeans in their own languages, like the Quechua manuscript written by an anonymous author in Huarochirf around 1608. Pre-Columbian Andean inscribed no words on paper, stone or wood. We can cull no words from the artifacts they left behind them. What they did leave behind is amazing quantities of marvelous cloth. Fiber arts have great time depth ( at least five millennia) in Andean civilization and must have been deeply meaningful, as they still are in parts of the rural Andes today. Not surprisingly, fiber was the medium in which Andeans developed techniques for recording information. By the Middle Horizon (c.1000AD) they were keeping records on complex sets of knotted strings. When Spaniards arrived in the early sixteenth century the lnkas were using these string devices to administer an extremely far-flung empire, stretching from southern Colombia to northern Chile. Spanish chroniclers were impressed with the amount, complexity, variety and reliability of information stored in these khipus (the word means knot in Quechua). Early-on the colonial regime allowed evidence from khipus (though not the khipus themselves) to be entered in court records and even required that native authorities keep tribute accounts on khipus. By late sixteenth century, however, Spanish inquisitors had caught on to another aspect of these string devices: A very un-Christian type of sacredness inhered in them; khipus recorded a complex system of shrines housing ancestral mummies, and some were even interred with them. Thus, khipus were outlawed as idolatrous; and khipu-users, if they persisted, did so at great risk. Standard histories assume that khipu-use died out at this point, and that by early seventeenth century khipus were a things of the past. Unfortunately the colonizing Spanish never figured out how the khipus actually worked; apparently the system was too foreign to learn. The lnka and pre-lnka khipus left to us-mostly looted and totally without provenance-are silent in our hands. No one has ever turned up a Rosetta Stone equivalent (as, for example, a khipu attached to a written account of its contents). Thus the Andean khipu remains a enigma. In its knotted strings we find a sophisticated communication device, developed in relative isolation from the rest of the world and apparently premised on principles radically different from those used in other forms of writing. Inka khipus are of a mind-boggling size and complexity that clearly goes beyond a knot-on-the-finger type of mnemonic device. Decoding them would not only to transform our understanding of one of the world's great (and least understood) empires, but could provide new perspectives on human intelligence as well. …

Unraveling Khipu's Secrets

Researchers are moving toward understanding the communicative power of khipu, the Inca9s enigmatic knotted strings, which wove an empire together. On page 1065 of this issue, researchers identify one way that data and instructions were passed up and down the Inca administrative hierarchy and tentatively identify the first known khipu word. Other researchers have independently announced a very early example of khipu; if verified, the find suggests that khipu were one of the world9s oldest complex recording systems.

Knotted self-dual strings coupled with Kalb-Ramond field

It is shown that knotted topological strings appear at the symmetric phase of Higgs field in the context of generalized Yang-Mills Chern-Simons Higgs model. The equation of motion of the knotted strings coupled with the Kalb-Ramond field is obtained. Based on the equation of motion, we found that there is a topological constrain on the coupling term between Maxwell field and the covariant derivative of Kalb-Ramond field, the topological number of this coupling is the topological number of the abelian Chern-Simons action on the knotted topological strings.

Jake Page's Indian Crime Fiction

Background of the half dozen most powerful authors of Native American crime fiction is Jake Page, who has published five novels on the subject so far: The Stolen Gods (1993), The Deadly Canyon (1994), The Knotted Strings (1995), The Lethal Partner (1997), and A Certain Malice (1998). His tribe of concentration is the Hopi of New Mexico and northeastern Arizona. He centers on their lives and activities around Santa Fe, New Mexico, a city for which he has a love-hate relationship - love of its beauty and tradition, contempt for the commercialization of its facade and its pretentiousness. Santa Fe, Page has his leading character say, has various social tribes that effectively define the and those tribes two overriding convictions. One is that Santa Fe is as close to the original Eden as mankind has come since Adam and Eve were banished (Gods 13). The second is that Santa Fe is a small city, with all of the advantages generally associated with a town of that size. Page thinks that both personal convictions about a geographical site may be pushed too far. His evaluation comes from a cool, dispassionate analysis, which reveals all kinds of serpents in the Garden, such as pride, greed, blatant ambition, desire to shine in the public eye, and self-nurtured ignorance. In other words, Santa Fe is or is coming to be like Albuquerque, which Page hates, and other American cities. Page, who has no Indian blood, writes of the Hopis of the area because he has lived and worked among them with his photographer wife, Susanne Anderson, since the early 1980s. He knows them well- not, he says in a letter to me, in the sappy way a lot of people are who meet Indians and want to wear feathers or whatever, but in a much more profound and respectful way. Knowledge breeds respect. He respects the Hopis, and they know that they can count on him for friendly assistance whenever he can help, as he has on several occasions. A Hopi tribal chieftain once asked Page if he publish magazine articles about the theft of four Indian gods, a matter of profound importance to the Hopis. Page wrote such an article for Smithsonian magazine, but it was turned down. Page feels, as he said in his biographical letter to me, the reason was that the article implicated a sister museum in the trading of Indian religious artifacts. This is a crime against the federal government because the tribes fall under the jurisdiction of the Bureau of Indian Affairs, and therefore are policed by the FBI. Page subsequently wrote another article for Connoisseur magazine, which also turned it down because, as Page told me in his biography, it would be offensive to galleries. Having been forced to drink the gall of the injustice of the dominant art world's treatment of less powerful cultures, Page decided to write a novel on the subject, focusing on the kachinas, the religious artifacts that he considers splendid and deserving of the praise voiced by some art historians who consider this art form the most profound in America. In his first novel, The Stolen Gods, Page shakes a large fist of accusation at art collectors and dealers who, by hook or crook, get control of and sell Hopi art to outside collectors who cannot understand and are indifferent to its religious significance. Callously and greedily they deprive the Indians of the soul of their existence, fattening on the commercial exploitation of religious artifacts. Page scornfully labels such practice simony, in the biblical sense of the word. Page does understand the significance of the kacbinas, their use in religious practices, and other manifestations of spirituality in the lifestyle of the Hopis. He wants to pay back the Indians for the sins of his fellow Americans and freely and fully offers his personal, family, and racial remorse and atonement for the misbehavior of fellow whites: Over the years, he says in the author's note to The Stolen Gods, many Hopi people have enriched my life and my family's lives in ways that one does not count. …

Anthropology. Cracking the khipu code.

Researchers are taking a fresh look at the cryptic Incan knotted strings known as khipu, which were long viewed simply as mnemonic devices or textile abacuses. Some are now suggesting that these complex textiles may have been a written language that used a binary code to store information. If so, this ancient three-dimensional writing system was unlike any other on Earth.

Quantization of energy and writhe in self-repellingknots

Probably the most natural energy functional to be considered forknotted strings is that given by electrostatic repulsion. Inthe absence of counter-charges, a charged, knotted stringevolving along the energy gradient of electrostatic repulsionwould progressively tighten its knotted domain into a point on aperfectly circular string. However, in the presence of chargescreening self-repelling knotted strings can be stabilized.It is known that energy functionals in which repulsive forcesbetween repelling charges grow inversely proportionally to thethird or higher power of their relative distance stabilizeself-repelling knots. Especially interesting is the case of thethird power since the repulsive energy becomes scale invariantand does not change upon Möbius transformations(reflections in spheres) of knotted trajectories. We observehere that knots minimizing their repulsive Möbius energyshow quantization of the energy and writhe (measure ofchirality) within several tested families of knots.

Localizationof breakage points in knotted strings

It is a common macroscopic observation that knotted ropes or fishing linesunder tension easily break at the knot. However, a more precise localizationof the breakage point in knotted macroscopic strings is a difficult task. Inthe present work, the tightening of knots was numerically simulated, acomparison of strength of different knots was experimentally performed and ahigh velocity camera was used to precisely localize the site where knottedmacroscopic strings break. In the case of knotted spaghetti, the breakageoccurs at the position with high curvature at the entry to the knot. Thislocalization results from joint contributions of loading, bending and frictionforces into the complex process of knot breakage. The present simulationsand experiments are in agreement with recent molecular dynamics simulations ofa knotted polymer chain and with experiments performed on actin and DNAfilaments. The strength of the knotted string is greatly reduced (down to 50%)by the presence of a knot, therefore reducing the resistance to tension of allmaterials containing chains of any sort. The present work with macroscopicstrings revels some important aspects, which are not accessible by experimentswith microscopic chains.

The dynamics of knotted strings attached to D-branes

We extend the general solution to the Cauchy problem for the relativistic closed string (Phys. Lett. B404 (1997) 57-65, hep-th/9704084) to the case of open strings attached to Dp-branes, including the cases where the initial data has a knotlike topology. We use this extended solution to derive intrinsic dynamical properties of open and closed relativistic strings attached to Dp-branes. We also study the singularity structure and the oscillating periods of this extended solution.

Stable knotted strings

We solve the Cauchy problem for the relativistic closed string in Minkowski space M 3+1, including the cases where the initial data has a knot like topology. We give the general conditions for the world sheet of a closed knotted string to be a time periodic surface. In the particular case of zero initial string velocity the period of the world sheet is proportional to half the length ( l) of the initial string and a knotted string always collapses to a link for t = l/4. Relativistic closed strings are dynamically evolving or pulsating structures in spacetime, and knotted or unknotted like structures remain stable over time. The generation of arbitray n-fold knots, starting with an initial simple link configuration with non zero velocity is possible.