Transcription
Most of us tie our shoelaces incorrectly. There are two primary methods to tie a knot in shoelaces: one involves going counterclockwise around the loop, while the other involves going clockwise. Although these two methods may look nearly identical, one of these knots is significantly superior to the other, as it does not loosen or come untied nearly as easily. To understand why this is the case, we must delve into knot theory, a branch of mathematics dedicated to identifying, categorizing, and comprehending every possible knot that could ever exist. So far, we have discovered 352,152,252 knots, each with its own distinct properties and characteristics.
The idea of having something akin to a periodic table for knots is fascinating, but knot theory extends beyond pure mathematics. It has proven remarkably useful in various fields, including the structure of proteins and DNA, the development of new materials potentially stronger than Kevlar, and the creation of life-saving medicines. This all stems from a quest to understand the seemingly humble knot.
So, what exactly is a knot? In everyday life, we encounter various types of knots; however, for rigorous study, knots must be pulled apart to analyze their composition. The challenge here is that knots are held together only by tension and friction, meaning that excessive force can cause them to fall apart. To facilitate the study of knots, mathematicians connected the two ends of the rope, allowing for the knot to be teased apart without fundamentally changing its structure. Thus, in knot theory, all knots exist on closed loops, making the simplest knot merely a circle, referred to as the "unknot."
Another example of a knot is the trefoil, which consists of a single piece of rope forming a closed loop. Two knots are considered different only if you cannot transform one into the other without breaking the loop. The trefoil is the simplest knot following the unknot because there is no way to convert it back into a circle without breaking it open, removing the knot, and then re-closing it, which results in two unknots. Distinguishing between two knots by sight can be surprisingly challenging. For instance, consider a mystery knot: is it an unknot, a trefoil, or neither? Upon closer inspection, one can untwist and rearrange it to reveal that it is indeed a trefoil.
It becomes evident that all too often, knots that appear complicated can, in fact, simply be unknots. The crux of the issue is that one cannot just randomly tangle some rope and connect the ends, as you need to prove that it is not simply a tangled version of another knot. This dilemma introduces the knot equivalence problem, a notoriously challenging question that has spurred the field of knot theory for over 150 years. Alan Turing highlighted this difficulty in his final publication, asserting that "no systematic method is yet known by which one can tell whether two knots are the same."
The Gordian knot, the most famous knot problem in history, serves as a historical reference. Legends claim that whoever could untangle this knot would rule all of Asia, and Alexander the Great is said to have sliced through it rather than unraveling it. In addition to the Gordian knot, other notable knots have persisted through history. The endless knot can be found in clay tablets from the Indus Valley and has appeared in Medieval Celtic designs as well as Chinese and Hindu contexts. The Incan civilization used knots on cords called quipu for tracking taxes and calendars. Even the House of Borromeo, an Italian noble family tracing its ancestry back to the 1300s, features the Borromeon rings in their coat of arms. The Borromeon rings represent a link, which is essentially a knot with multiple loops of rope.
The knot equivalence problem only garnered attention centuries later, in January 1867, when Scottish physicist Peter Guthrie Tait unveiled his homemade smoke machine to the renowned scientist William Thomson, later known as Lord Kelvin. Tait was intrigued by a paper asserting that a vortex ring should exhibit eternal stability in an ideal fluid. Consequently, he set up two wooden boxes containing a toxic mixture of ammonia, sulfuric acid, and salt. By tapping a towel stretched across each box, Tait produced chemical smoke that escaped through a circular cutout in perfect rings.
Fascinated, Kelvin pondered the composition of atoms, a fundamental question of his time, and suddenly saw an answer. He proposed that atoms must be composed of vortex rings of an invisible medium called ether. He theorized that different knots of vortex rings corresponded to different elements, where the shape of the Hopf link explained the double spectral lines of sodium, while the simple unknot ring represented hydrogen. Although Tait was skeptical, Kelvin's vortex model of the atom became a leading theory, prompting Tait to investigate knots more seriously, envisioning a periodic table of elements represented by the knots he discovered.
To categorize knots, Tait employed a concept known as the crossing number. This involves analyzing the simplest form of a knot—one without extraneous twists or tangles—and counting all its crossings. Through laborious research, Tait discovered numerous knots, including a three-crossing knot (the trefoil), a four-crossing knot (the figure eight), and an array of other crossings. It's essential to note that knots can be additive, permitting the combination of several knots into a new composite knot. However, some knots cannot be decomposed into simpler knots, earning them the classification of prime knots.
Tragically for Tait, red flags began to appear regarding Lord Kelvin's vortex theory. Dmitri Mendeleev's first periodic table was published in 1869, and the Michelson-Morley experiment in 1887 raised doubts concerning the existence of ether. The most disheartening revelation came in 1897 when J.J. Thomson discovered the electron, revealing that particles existed within atoms and were smaller than the atom itself. Nevertheless, Tait remained immersed in his knot research, enlisting the help of his academic rival and close friend, James Clerk Maxwell, who would be a knot enthusiast for the remainder of his life.
With Tait's encouragement, Maxwell ultimately published a series of letters leading to Tait's list of knots totaling seven crossings in 1877, marking the first mathematics paper with the term "knots" in its title. However, Tait paused his search for seven years, stating, "The requisite labor increases with extreme rapidity as the number of crossings is increased," and invited others with "the requisite leisure" to extend the list, ideally up to eleven crossings.
Answering Tait's call, two mathematicians—Thomas Kirkman and Charles Little—joined forces with Tait, leading to the discovery of all 21 eight-crossing knots, 49 nine-crossing knots, and 166 ten-crossing knots by 1899, just two years before Tait's passing. This meticulous process involved painstaking manual labor, which Tait acknowledged in his paper, admitting, "I cannot be absolutely certain that all those groups are essentially different, one from another." During his work, Tait, Kirkman, and Little stumbled upon the knot equivalence problem, which remained crucial to understanding how to tell knots apart.
Inexplicably, their tabulation of knots stood unchanged for 75 years until a single correction was made in 1973. Notably, little progress was accomplished on the knot equivalence problem following Tait's death until 1927, when German mathematician Kurt Reidemeister proposed a radical theorem indicating that only three types of moves are necessary to transform any two identical knots into each other. These moves include the twist, the poke, and the slide, allowing knot theorists to prove the identity of some knots. However, the challenge of proving whether two knots differ remains daunting; Reidemeister moves can be applied indefinitely to one knot without it resembling the other, allowing for the possibility that they may be identical.
This difficulty likely influenced Turing's characterization of the knot equivalence problem as potentially undecidable. Nonetheless, in 1961, mathematician Wolfgang Haken developed a computer algorithm that resolved the knot equivalence problem definitively for the specific case of distinguishing any knot from the unknot. However, Haken’s solution was lengthy, with a paper nearly 130 pages long, and the algorithm required more time to execute than the age of the universe would permit for larger knots.
In 2001, advancements built upon Haken's work allowed mathematicians to distinguish between any knot and the unknot by establishing an upper bound on the number of Reidemeister moves necessary to connect them. By examining all sequences of Reidemeister moves up to this number, researchers could determine if the knot was an unknot or not. However, problems arose, as the upper bound set was two to the power of 100 billion n moves. As of today, this upper bound has improved significantly to just 236 n to the power of 11. Despite being more manageable, this bound is still astronomical, as checking all possible sequences of Reidemeister moves up to this limit exceeds the number of stars in the observable universe.
In 2011, mathematicians established an upper bound on the number of Reidemeister moves necessary to connect any two knots or links, thereby solving the entire knot equivalence problem. This upper bound is substantial: it begins with raising two to the second power, then two again, and continues this process while raising two to itself 10 to the million n times, culminating with n again. This represents one of the largest numbers conveyed publicly in a video.
The challenge behind telling two knots apart can lead to surprising results, evidenced by the fact that 350 million distinct knots have been categorized. Some characteristics of knots, labeled invariants, remain constant regardless of how much they twist or tangle. Although these invariants are not perfect discriminators, they can serve as identifiers for particular knots. The crossing number is one such invariant—two knots cannot be identical if they possess different crossing numbers, yet calculating crossing numbers can be surprisingly complicated.
Additional crossings can be easily incorporated into any knot, and variations of the same knot can be represented as different projections. The crossing number effectively measures the least number of crossings within a knot's simplest projection, known as its reduced form, but ensuring a knot is fully reduced poses challenges. Instead, knot theorists can utilize other invariants valid for all projections of a knot. The first such invariant is tricolorability, which assesses whether a knot can be colored with three different colors. A diagram of a knot can be analyzed; segments are colored, separated by undercrossings where movement is suspended.
Tricolorability adheres to two rules: at least two colors must be used (one color could apply to any knot), and at crossings, the three strands intersecting must either be identical in color or completely distinct (no two colored strands at a crossing). Knots are categorized simply as either tricolorable or non-tricolorable. If two knots differ in this attribute, they are not identical. Remarkably, tricolorability persists across any projection of the same knot.
While it appears outlandish that tricolorability is invariant, it can be proven considering Reidemeister moves. The twist maneuver preserves the color; the poke creates a third color at each intersection; the slide maintains the existing three-color structure. Thus, tricolorability remains unaffected by any Reidemeister moves executed.
We can now illustrate the difference between the trefoil and the unknot in terms of tricolorability. The unknot cannot be tricolored due to the inability to utilize at least two colors, while trefoil easily accommodates three distinct colors across its segments. Consequently, since every projection of the trefoil proves tricolorable, while the unknot does not, it is clear that the trefoil and unknot are, in fact, distinct knots.
Despite tricolorability's limitations in specificity—it only provides a binary classification across knots—progressing to a more extensive invariant yields p-colorability, where p represents any prime integer aside from two. This approach assigns numerical values to each strand, adding complexity to the invariants, with two primary rules established. First, at least two different numbers must be utilized, and second, the addition of the two lower strands divided by p must yield the same remainder as the top strand divided by p.
Expanding on the figure-eight knot, the transitions from three-colorability to five-colorability using integers reveals that it fits the criteria, showing it is different from the uncolorable unknot. While p-colorability stands as a powerful tool—demonstrating that the unknot is devoid of colorability, thus differentiating it from all knots with any degree of colorability—it does not encompass every potential knot.
Some of the most potent invariants currently available for distinguishing unique knots are polynomials. The Alexander polynomial, the first of its kind discovered back in 1923, relies on two primary rules: the Alexander polynomial of the unknot equals one, and one can zoom in on any single crossing of a knot to examine three possible positions: forward, backward, and separate. The polynomial interprets the relationships between the three resulting knots.
Let’s calculate the Alexander polynomial for the unlink: if we closely analyze the separate crossing, both resulting knots become unknots. Thus, it follows that the Alexander polynomial for the unlink must equal zero. The same logic can apply to other knots, such as the Hopf link and trefoil, yielding different polynomials. After enduring as the primary knot invariant for over 60 years, the Alexander polynomial was challenged in 1984 by an unexpected discovery.
Mathematician Vaughan Jones, who was working on statistical mechanics, realized that his equations resembled those in knot theory. With the assistance of knot theorist Joan Birman at Columbia University, he refined these equations into a brand-new polynomial invariant. This new realization, known as the Jones polynomial, pairs similarity with improved specificity in distinguishing numerous knots. For this finding, Jones received the Fields Medal in 1990.
The introduction of the Jones polynomial ignited excitement within the knot theory community. Almost immediately, six mathematicians independently developed enhanced versions of Jones' polynomial, culminating in the HOMFLY polynomial, and subsequently the HOMFLY-PT polynomial. No single invariant suffices to work alone in the quest to uniquely identify knots. Instead, employing various invariants alongside Reidemeister moves enables knot theorists to tackle their task from two distinct angles, ultimately aiming to differentiate every single knot.
Yet, the endeavor is not flawless. An instance arose when Kenneth Perko, a lawyer with a background in knot theory, discovered two knots previously listed next to each other in Tait's tables. Though recorded as unique for over 75 years, he identified that they were the same knot through Reidemeister moves. The discovery, known as the Perko pair, resulted in a correction to Tait's table, reducing the number of recorded ten-crossing knots from 166 to 165.
Despite the arduous task of cataloging knots up to ten crossings, no one ventured to address eleven crossings until John Conway, who claimed to have identified all 552 in just an afternoon. This achievement marked the last instance of tabulation by hand, as subsequently, computer algorithms emerged to count knots through 12 and 13 crossings. Researchers Dowker, Thistlethwaite, Hoste, and Weeks later collaborated to document all knots through 14, 15, and 16 crossings in a paper titled "The First 1,701,936 Knots."
Utilizing invariants, the method developed in their work remains the standard in contemporary knot tabulation, systematically listing all possible knots to eliminate duplicates. In 2020, mathematician Ben Burton single-handedly tabulated all 17, 18, and 19 crossing knots, bringing the total known prime knots to 352,152,252. The computationally intensive project required several hundred computers running for months to achieve the final tally.
The intense difficulty of accurately counting every knot and removing duplicates poses a significant challenge. However, for mass-generating distinct knots, one can construct alternating knots, where crossings consistently alternate between over and under. This method is much simpler but excludes most unique knots. In 2007, this process was employed to uncover alternating knots reaching a staggering 24 crossings, leading to a cumulative total of 159,965,097,353 known knots.
Initially, knot theory existed as a purely mathematical interest, with algorithms, invariants, and tabulations serving as knowledge sought for its own sake. Yet, in 1989, French chemist Jean-Pierre Sauvage achieved a groundbreaking milestone by creating the first synthetic knotted molecule by tying molecules around copper ions to form a trefoil knot. This synthetic knot prevents the atoms from unraveling, trapping them in higher energy states that grant different properties.
As demonstrated, any knot tied into a molecule alters its properties, presenting the potential for over 159 billion new unique materials from just a single molecule. However, chemists have only succeeded in creating five additional molecular knots since the trefoil, as constructing these knots is tremendously challenging. Due to the necessity of precise arrangements, chemists must design molecules that self-assemble into these knots. Knot theory further assists by identifying knots that could correspond with available molecular templates, as symmetric knots are easier to form.
Currently, the most intricate knot created is the 819 knot, consisting of 192 atoms tied around a central chloride ion. This molecule has gained the Guinness World Record for being the tightest knot in the world, defined by its crossings per unit length, in this case, eight crossings in just 20 nanometers. Interestingly, since this knot is knotted around a chloride ion, once that ion is removed, the resulting molecule stands as one of the strongest chloride binders known.
Though still in its infancy regarding specific applications, knot theory is crucial for biological processes responsible for saving millions of lives. Bacterial DNA consists of a single loop of double helix, causing it to form a knotted link whenever it replicates. The tangled DNA cannot separate into two cells without appropriate untying. Bacteria utilize an enzyme known as type two topoisomerase to snip and reconnect the DNA, converting linked DNA back into an unlink, allowing for clean replication.
Inhibiting type two topoisomerases inhibits bacterial replication, leading to death. This mechanism underscores how various antibiotics operate, especially quinolones, which target the enzyme. Although human DNA is not circular, its extensive length—two meters per cell—can also result in tangles. As such, each cell contains enough DNA that, if packed, equates to roughly 200 kilometers of fishing line crammed inside a basketball.
When tangles naturally arise, human type two topoisomerases intervene to manage crossings, facilitating the resumption of activities. While the human enzymes operate differently from their bacterial counterparts, they remain significantly affected by inhibition. Specifically, this inhibition is utilized during chemotherapy, targeting rapidly dividing cancer cells. Biologists employed knot theory to decode the mechanism behind type two topoisomerases, determining that it decreases the crossing number of knots in DNA by two at a time by cutting and rejoining entire double strands.
Beyond DNA, one percent of all proteins also exhibit various knots within their structures. A misnotted protein can malfunction, revealing the importance of accurately distinguishing knots to better understand the mechanisms involved and how to repair or capitalize on them. In practical terms regarding shoelaces, both common knot-tying methods create two trefoils stacked on top of each other.
Going counterclockwise around the loop results in two identical trefoils, known as a granny knot. Conversely, performing the clockwise operation yields mirrored trefoils, recognized as a square knot, which does not loosen as easily. Thus, it is advisable for everyone to tie their shoelaces in this clockwise manner—even though many, including myself, typically do not.
Simple overhand knots equate to a trefoil. Additionally, the bowline knot, commonly used for boating or securing objects, represents the six-two knot, while any knot tied without using the ends (also known as "in the bite") is simply an unknot, as seen with slipknots.
In 2007, researchers Dorian Raymer and Douglas Smith conducted 3,415 trials spinning string in various boxes to understand how knots form in real-world contexts. They successfully generated 120 knot types, some comprising 11 crossings. Their findings indicated that longer agitation durations increased the likelihood of knotting. The presence of longer strings also raised this probability, albeit diminishing when the string was confined within smaller boxes that restricted its motion.
To prevent items like headphones from knotting in your pocket—especially when constraints of string length or agitation duration come into play—experimenters advise confining them to a small space. Raymer and Smith proposed a model for real-world knot formation, stating that loops emerge first when a string is placed in a container. When agitated, the unanchored string's end weaves through these loops, ultimately braiding itself into knots.
As an observation, coiling wires may inadvertently lead to failure, creating several loops through which the loose end can braid naturally into a knot. Therefore, restricting the wire’s movement is key, whether by using a small container or increasing the stiffness of the string. Supercoiling the DNA demonstrates a similar principle, and you can replicate this effect with your wires by doubling them up and twisting the center. While the end product may appear tangled, simply pulling the opposite ends apart results in no knots whatsoever.
Raymer and Smith's study garnered an Ig Nobel Prize and served as an important citation for analysis in studies concerning knots in surgical catheters. Their work has even been linked to a patent developed by Apple for stiffer earbud wires. Knot theory originated as a pursuit for a comprehensive understanding of existence but has since evolved into a vital field, intersecting various disciplines from headphone tangles to materials science and chemotherapy.
In 1889, Lord Kelvin delivered a presidential address to the British Institution of Electrical Engineers, reflecting on his unfulfilled theories regarding atomic knots. “I am afraid I must end by saying that the difficulties are so great in the way of forming anything like a comprehensive theory that we cannot even imagine a finger-post pointing to a way that leads us towards the explanation… But this time next year, this time 10 years, this time 100 years…”
He expressed confidence that concepts now shrouded in mystery would eventually be deciphered, unveiling a clear path toward understanding the universe. Knot theory serves as a perfect illustration of how knowledge from one area can evolve into a tool for comprehending an array of others. From discerning a trefoil from an unknot, knot theorists have significantly advanced, culminating in discoveries of new proteins.
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