Knot Theory Quotes

Perfection itself is a flaw, an odd knot in the cosmic fabric of evenly-braided imperfections.

Yarn is fundamental, knots are not. Similarly, Hilbert and Mike envisioned a natural order in which the geometry of fields is foremost and twists manifest themselves as particles.

He moved with a slow gait. I never saw him dance. He was a man who wrote, who interpreted the world. Wisdom grew out of being handed just the smallest sliver of emotion. A glance could lead to paragraphs of theory. If he witnessed a new knot among a desert tribe or found a rare palm, it would charm him for weeks. When we came upon messages on our travels – any wording, contemporary or ancient, Arabic on a mud wall, a note in English written in chalk on the fender of a jeep – he would read it and then press his hand upon it as if to touch its possible deeper meanings, to become as intimate as he could with the words.

To test this theory, I sent this picture to my mother and asked her what she thought had happened. She immediately replied,2 “The kid knocked over the vase and the cat is investigating.” She cleverly rejected alternate hypotheses, including: The cat knocked over the vase. The cat jumped out of the vase at the kid. The kid was being chased by the cat and tried to climb up the dresser with a rope to escape. There’s a wild cat in the house, and someone threw a vase at it. The cat was mummified in the vase, but arose when the kid touched it with a magic rope. The rope holding the vase broke and the cat is trying to put it back together. The vase exploded, attracting a child and a cat. The child put on the hat for protection from future explosions. The kid and cat are running around trying to catch a snake. The kid finally caught it and tied a knot in it.

People make the mistake of supposing that genius is complicated. It is the opposite. We regular folks are complicated — tied in knots of ambivalence and befogged with uncertainties. Genius has the economy of a machine with a minimum of moving parts. Everything about Picasso came to bear when he drew a line.

I want to quote that poem in something I'm writing," he explained, "and can you tell me the last line of it ? " Lou answered mechanically, as if he had pressed a button: "Death is not a way out of it!" "A very strange theory, that about death," he said. "I wonder if there's anything in it. It would really be too easy if we could get out of our troubles in so simple a fashion. It has always seemed to me that nothing can ever be destroyed. The problems of life are really put together ingeniously in order to baffle one, like a chess problem. We can't untie a real knot in a closed piece of string without the aid of the fourth dimension; but we can disentangle the complexities caused by dipping the string in water-and such things," he added, with an almost malicious gravity in his tone. I knew what he meant. " It might very well be," he continued, " that when we fail to solve the puzzles of life, they remain with us. We have to do them sooner or later ; and it seems reasonable to suppose that the problems of life ought to be solved during life, while we have to our hands the apparatus in which they arose. We might find that after death the problems were unaltered, but that we were impotent to deal with them. Did you ever meet any one that had been indiscreet about taking drugs ? Presumably not. Well, take my word for it, those people get into a state which is in many ways very like death. And the tragic thing about the situation is this ; that they started taking the drugs because life, in one way or another, was one too many for them. And what is the result ? The drugs have not in the least relieved the monotony of life or whatever their trouble was, and yet they have got into a state very like that of death, in which they are impotent to struggle. No, we must conquer life by living it to the full, and then we can go to meet death with a certain prestige. We can face that adventure as we've faced the others.

Twisting space-time into knots requires energy on a scale that will not be available within the next several centuries or even millenia-if ever. Even if all the nations of the world were to band together to build a machine that could probe hyperspace, they would ultimately fail. And, as Guth points out, the temperatures necessary to create a baby universe in the laboratory is 1,000 trillion trillion degrees, far in excess of anything available to us. In fact, that temperature is much greater than anything found in the interior of a star. So, although it is possible that Einstein's laws and the laws of quantum theory might allow for time travel, this is not within the capabilities of earthlings like us, who can barely escape the feeble gravitational field of our own planet. While we can marvel at the implications of wormhole research, realizing its potential is strictly reserved for advanced extraterrestrial civilizations.

And here she was, in the middle of a tangled situation she didn't remember or understand, painfully aware that a killer or killers on the loose almost certainly knew a hell of a lot more about what was going on than she did. As independent and self-reliant as she was, Riley had never felt so unsure of herself. She was adept at role-playing-it was one of her strengths-but this? This was a very, very dangerous game of blind man's bluff, and the one wearing the blindfold-her-had cotton in her ears and a clothespin on her nose as well. With the exception of Gordon, she didn't know who to trust, and he could offer little more than moral support since, if she had even reached any conclusions or formed any theories since arriving here, she had not confided them to him. As for the other man she was intimately close to… "Riley? Ready to order?" She looked across the top of her menu at this pale-eyed stranger whose bed she apparently shared, and ignored the cold knot in the pit of her belly to say calmly, "I'm ready." It was the second time she'd said that in the last couple of hours. She only hoped it was true.

All sex is weird in theory, but you like someone and so you give it a shot, and then maybe it’s good or intriguing enough you do weird things again. Sometimes, those weird sex things become great and then you get used to it. And, sometimes it’s genuinely just not what someone wants.

Comics ... are sometimes four-legged and sometimes two-legged and sometimes fly and sometimes don't ... to employ a metaphor as mixed as the medium itself, defining comics entails cutting a Gordian-knotted enigma wrapped in a mystery ...

Joining the world of shapes to the world of numbers in this way represented a break with the past. New geometries always begin when someone changes a fundamental rule. Suppose space can be curved instead of flat, a geometer says, and the result is a weird curved parody of Euclid that provides precisely the right framework for the general theory of relativity. Suppose space can have four dimensions, or five, or six. Suppose the number expressing dimension can be a fraction. Suppose shapes can be twisted, stretched, knotted. Or, now, suppose shapes are defined, not by solving an equation once, but by iterating it in a feedback loop. Julia, Fatou, Hubbard, Barnsley, Mandelbrot-these mathematicians changed the rules about how to make geometrical shapes. The Euclidean and Cartesian methods of turning equations into curves are familiar to anyone who has studied high school geometry or found a point on a map using two coordinates. Standard geometry takes an equation and asks for the set of numbers that satisfy it. The solutions to an equation like x^2 + y^2 = 1, then, form a shape, in this case a circle. Other simple equations produce other pictures, the ellipses, parabolas, and hyperbolas of conic sections or even the more complicated shapes produced by differential equations in phase space. But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, dynamic instead of static. When a number goes into the equation, a new number comes out; the new number goes in, and so on, points hopping from place to place. A point is plotted not when it satisfies the equation but when it produces a certain kind of behavior. One behavior might be a steady state. Another might be a convergence to a periodic repetition of states. Another might be an out-of-control race to infinity.

Braid groups have many important practical applications. For example, they are used to construct efficient and robust public key encryption algorithms.7 Another promising direction is designing quantum computers based on creating complex braids of quantum particles known as anyons. Their trajectories weave around each other, and their overlaps are used to build “logic gates” of the quantum computer.8 There are also applications in biology. Given a braid with n threads, we can number the nails on the two plates from 1 to n from left to right. Then, connect the ends of the threads attached to the nails with the same number on the two plates. This will create what mathematicians call a “link”: a union of loops weaving around each other. In the example shown on this picture, there is only one loop. Mathematicians’ name for it is “knot.” In general, there will be several closed threads. The mathematical theory of links and knots is used in biology: for example, to study bindings of DNA and enzymes.9 We view a DNA molecule as one thread, and the enzyme molecule as another thread. It turns out that when they bind together, highly non-trivial knotting between them may occur, which may alter the DNA. The way they entangle is therefore of great importance. It turns out that the mathematical study of the resulting links sheds new light on the mechanisms of recombination of DNA. In mathematics, braids are also important because of their geometric interpretation. To explain it, consider all possible collections of n points on the plane. We will assume that the points are distinct; that is, for any two points, their positions on the plane must be different. Let’s choose one such collection; namely, n points arranged on a straight line, with the same distance between neighboring points. Think of each point as a little bug. As we turn on the music, these bugs come alive and start moving on the plane. If we view the time as the vertical direction, then the trajectory of each bug will look like a thread. If the positions of the bugs on the plane are distinct at all times – that is, if we assume that the bugs don’t collide – then these threads will never intersect. While the music is playing, they can move around each other, just like the threads of a braid. However, we demand that when we stop the music after a fixed period of time, the bugs must align on a straight line in the same way as at the beginning, but each bug is allowed to end up in a position initially occupied by another bug. Then their collective path will look like a braid with n threads. Thus, braids with n threads may be viewed as paths in the space of collections of n distinct points on the plane.10

Looked at differently, these various levels are possibilities of “bodily” existence. Depending on which “body,” or state of existence, we set our heart on, we will obtain that very form of embodiment once we have quit our present physical vehicle. Thus we may resume a human form, become an angelic being (deva) inhabiting the subtle realm, sink into the cosmic matrix itself as a prakriti-laya, or recover our true identity as the Self in the blissful “company” of the Divine Person, Lord Krishna. The phrase “to set one’s heart on something” is perhaps more expressive of the intended meaning of “remembering” than is “thinking.” The latter has too abstract a flavor, while the former is indicative of a deep-level process. This becomes clear when one realizes that one of the synonyms of “meditation” is in fact “remembering.” Meditative remembering means sending taproots down into the hidden recesses of our being, into the depth-mind (buddhi). The depth-mind is the storehouse of the essence (i.e., karmic deposits) of our world experience. It has been likened to a net whose knots are the impressions left behind by our volitional activity. Meditation is a partial enactment of the process of dying. Conversely, death is the meditative process taken to its logical conclusion. At death, the mind disentangles itself from the physical body, and the center of identity is shifted to the subtle vehicle—the so-called astral body or “subtle vehicle” (sūkshma-sharīra). The quintessence of the contents of the mind, in the form of the subliminal impressions (samskāra) stored away in the depth-mind, is the factor that determines the after-death fate of the deceased according to the iron-law of moral retribution or karma.

The thickness of the armor decking was increased by 1.75 inches, and giant torpedo bulges, or blisters, were added below the waterline on the exterior of each side of the hull. In theory, these bulges provided some measure of protection from torpedoes or near-miss bombs. They were filled with air in the outer half and water in the half next to the hull, and were designed to absorb the shock of an explosion and dissipate potential damage to the ship. Bulkheads bisecting the bulge limited flooding and damage to a particular section. All this meant more weight and new boilers and turbines were also installed to keep up Arizona’s speed, which peaked at 20.7 knots (23.8 mph) during post-overhaul sea trials. Returned to full commission on March 1, 1931, Arizona embarked President Herbert Hoover from Hampton Roads, Virginia, for a ten-day tour of the Caribbean, calling at Puerto Rico and the Virgin Islands. It was the sort of low-key inspection of naval operations combined with a warm respite from wintry Washington that American presidents happily undertook in those years. Franklin Roosevelt would soon become a master of it.

This view, which I call the theory of constructed emotion, offers a very different interpretation of the events during Governor Malloy’s speech. When Malloy’s voice caught in his throat, it did not trigger a brain circuit for sadness inside me, causing a distinctive set of bodily changes. Rather, I felt sadness in that moment because, having been raised in a certain culture, I learned long ago that “sadness” is something that may occur when certain bodily feelings coincide with terrible loss. Using bits and pieces of past experience, such as my knowledge of shootings and my previous sadness about them, my brain rapidly predicted what my body should do to cope with such tragedy. Its predictions caused my thumping heart, my flushed face, and the knots in my stomach. They directed me to cry, an action that would calm my nervous system. And they made the resulting sensations meaningful as an instance of sadness.

According to an esoteric explanation, the Sanskrit term mantra signifies “that which protects (trāna) the mind (manas). Specifically, mantra is a sound (letter, syllable, word, or phrase) that is charged with transformative power, such as the letter a, the sacred syllable om, the word hamsa, or the phrase om mani padme hūm. Thus a mantra could be explained as a potentized sound by which specific effects in consciousness can be produced. Most high-minded practitioners are reluctant to use mantras for anything other than the greatest human goal (purusha-artha, written purushārtha), which is liberation. In Tantric rituals, mantras are used to purify the altar, one’s seat, implements such as vessels and offering spoons, or the offerings themselves (e.g., flowers, water, food), or to invoke deities, protectors, and so on. Yet, the science of sacred sound (mantra-shāstra) has since ancient times been widely put to secular use as well. In this case, mantras assume the character of magical spells rather than sacred vibrations in the service of self-transformation and self-transcendence. The serpent energy hidden in the body is associated with the Sanskrit alphabet constituted of fifty basic letters, or sound vibrations, which go into the making of mantras. In contrast to ordinary words, however, mantras most often do not have a particular meaning, and their potency is tapped into through frequent repetition, whether mentally, whispered, or aloud. It is not commonly understood that for a sound to be a mantra, it must have been given in the context of initiation (dīkshā), whether formally or informally. Only then does the mantra have truly transformative power. For a mantra to become “active” or “awakened,” it must be recited at least 100,000 times. A mantra lacking in “consciousness” is just like any other sound. As the Kula-Arnava-Tantra (15.61–64) states: Mantras without consciousness are said to be mere letters. They yield no result even after a trillion recitations. The state that manifests promptly when the mantra is recited [with “consciousness”], that result is not [to be gained] from a hundred, a thousand, a hundred thousand, or ten million recitations. O Kuleshvarī, the knots at the heart and throat are pierced, all the limbs are invigorated, tears of joy, gooseflesh, bodily ecstasy, and tremulous speech suddenly occur for sure . . . . . . when a mantra endowed with consciousness is uttered even once. Where such signs are seen, that [mantra] is said to be according to tradition. Mantras of concentrated potency are known as “seed syllables” (bīja). Om is the original seed syllable, the source of all others. The Mantra-Yoga-Samhitā (71) calls it the “best of all mantras,” adding that all other mantras receive their power from it. Thus om is prefixed or sometimes also suffixed to numerous mantras, such as om namah shivāya

The next hurdle is the recognition that we have many deeply ingrained habit patterns that take time—a lot of time—to change. At first the typical neophyte is sure that he or she has a tremendous capacity and will grow more quickly than others. Then the sobering realization dawns that the degree of self-transformation is equal to the effort made. If neophytes have persisted thus far, they will almost inevitably encounter doubt (samshaya)—doubt about their own capacity; doubt about their teacher; doubt about the efficacy of the teaching. It is not far from the truth to say that practitioners who do not befriend doubt are bound to become self-deluded. If there really is no doubt or self-delusion, then they are quite simply enlightened. Another obstacle, not often identified, is the fact that practitioners’ karmic tendencies (read unconscious or semiconscious habit patterns) are magnified because awareness is enhanced through regular practice. This can be likened to a bright searchlight shining deep into the well of the mind. In the depth of the unconscious reside all kinds of unpleasant realities that get flushed out by steady application to self-inspection and self-understanding. At times, the unconscious materials that drift into the conscious mind seem overwhelming, and then it becomes clear that spiritual life is a form of brinkmanship. The Indic tradition speaks of the razor-edged path. Gradually spiritual practitioners learn to overcome their intrinsic materialism (i.e., constantly thinking in terms of the visible reality only). There is a progressive loosening of the ego knot or “self-contraction” (ātma-samkoca) by which the ordinary individual anxiously seeks to hold everything together. Spiritual practitioners learn to be humorous about everything, including themselves. Life is seen from a new perspective: as a strange play in which we are willy-nilly involved and which we can either misunderstand and suffer or understand and transcend even while being fully engaged in its drama. Practitioners must prevail over spiritual materialism—the false sense of accumulating “higher” experiences. They can realize inner freedom only to the extent that even the goal of liberation is renounced. Liberation, or enlightenment, is not a thing to be attained or acquired. It is living in the moment from the most profound understanding and without egoic attachment to anything. Those who parade their extraordinary spiritual accomplishments in front of others are possibly the least illumined of all. They merely substitute material commodities for “spiritual” merchandise. The Indic heritage knows of many adepts who after years of intense practice achieved a high state of consciousness or astounding paranormal ability only to promptly plunge from grace. The higher the adept’s elevation, the steeper the drop into oblivion and misery. Therefore the authorities of Yoga ever admonish practitioners to be circumspect, to keep their attainments to themselves, to focus on the cultivation of moral integrity, understanding, self-transcendence, and not least service to others.

How will that infrastructure, that secret of secrets this side of our theses and our theory, be able in turn to rest upon the acts of absolute consciousness? Does the descent into the realm of our 'archeology' leave our analytical tools intact? Does it make no changes at all in our conception of noesis, noema, and intentiomality--in our ontology? After we have made this descent, are we still entitled to seek in an analytics of acts what upholds our own and the world's life without appeal? We know that Husserl never made himself too clear about these questions. A few words are there like indicators pointing to the problem--signaling unthought-of elements to think about. To begin with, the element of a 'pre-theoretical constitution,' which is charged with accounting for 'pre-givens,' those kernels of meaning about which man and the world gravitate. We may with equal truth say of these pre-givens (as Husserl says of the body) either that they are always 'already constituted' for us or that they are 'never completely constituted'—in short, that consciousness is always behind or ahead of them, never contemporaneous. Husserl was undoubtedly thinking of these singular beings when in another connection he evoked a constitution which would not proceed by grasping a content as an exemplification of a meaning or an essence (Auffassungsinhalt-Auffassung als . . .) , an operating or latent intentionality like that which animates time, more ancient than the intentionality of human acts. There must be beings for us which are not yet kept in being by the centrifugal activity of consciousness: significations it does not spontaneously confer upon contents, and contents which participate obliquely in a meaning in the sense that they indicate a meaning which remains a distant meaning and which is not yet legible in them as the monogram or stamp of thetic consciousness. In such cases we do still have a grouping of intentional threads around certain knots which govern them, but the series of retro-references (Rückdeutungen) which lead us ever deeper could not possibly reach completion in the intellectual possession of a noema. There is an ordered sequence of steps, but it is without end as it is without beginning.

ALTRUISM WAS, from the very start, a problem for the Darwinian theory of evolution, if not something worse than a problem. As a result, Darwinians have always been under a certain temptation to "cut the knot," and deny the very existence of altruism. This