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Hello, good evening. Good afternoon. Good day, wherever you are in the world and welcome to the Black Belt Academy of Surgical Skills. My name is David o'regan. I'm a professor in the Medical Education Research and Development Unit of the Faculty of Medicine at the University of Malaya. I'm also the immediate past director of the Faculty of Surgical Trainers of the Royal College of Surgeons of Edinburgh, who accredit this course, we would not be able to reach 133 countries without the help and support of me, the platform that you have signed on. And this evening, our producer is Vanish and he is 1/4 year medical student here in the Faculty of Medicine. 24 countries have registered this evening from Australia, Bulgaria, China, Egypt, Iraq, Libya, Mexico, Sri Lanka, Sudan, Malaysia, Thailand, Ukraine, United Arab Emirates, United Kingdom and the USA. This is your first time joining us. Welcome. And if you're returning, thank you very much and continue to pass the word around. We're very excited at the Black Belt Academy because my fellow sense in the United Kingdom, Chris and John and Sys Michael together with Alice and I will be attending the Association and Surgeons in training, meeting in Belfast and running a walk in skills workshop. And I'm joining a fellow Louis Philippe online for a major conference in Bogota in Colombia. And we will be teaching the surgical principles. Tonight, we're going to talk about knots and we know you can get knots in muscles, knots in woods and there knots is a measure of speed in aviation and on the water. But actually a knot is a condition that is inflicted on a long flexible object. For example, you can get a knot in your hair and knot may be regarded as an interlacing or looping of string in any fashion. And then joining the ends. Not theory in mathematics is the study of closed curves in three dimensions and their possible deformations without one part cutting through another. The first question that arises is whether one such curve is truly knotted or can simply be entangled, that is whether or not excuse the pun, no one can deform it in a space into a standard unnoted curve like a circle. The second question the mathematicians ponded is whether more generally given curves, whether two given curves represent different knots or are really the same knot in the sense that they one can be continuously deformed into another, perhaps like isomers. The basic tool for classifying knots consists of projecting a knot onto a plane. So picture the shadow of a knot under a light And then what you do is actually count the number of times the each knot crosses, whether it's over or whether it's undone. And the complexity of a knot is the least number of crossings that occur as is not as moved around in all possible ways. And the simplest and possible true knot is a true foil knot or an overhand knot which has three crossings. And the order of this knot is therefore denoted as three. Even the simple note has two configurations that cannot be deformed into one another. They're like mirror images and there are no knots with fewer crossings and all others have at least four. Now, the number of distinguishable knots increases rapidly as the order of these crossings increases. For example, there are almost 10,000 distinct knots with 13 crossings and over a million with 16 crossings. And that's the highest known by the end of the 20th century. And certain order of higher knots cannot be resolved into combinations. And these combinations or products are I suppose are factors of low order knots. For example, the square knot and the granny knot are six order knots and are products of two tree foils that are the same or opposite. In 100 as I said, like isomers, not that cannot be resolved are called prime knots. And this deviating a little bit may be true of the mythical Gordon not you see. In 333 BC, the Fijians were without a king but an oracle by Teleus, the ancient capital of la decreed that the next man to enter the city driving an ox cart should become their king lo and behold a peasant farmer named Gordius drove into the town with an ox cart and was immediately declared king out of gratitude. His son Midas dedicated the ox card to the Fi God s or otherwise zeal and tied it to a post with an intricate knot of coral bark. This became the legendary Gordian knot and reputedly whoever could untired would then be destined to rule the whole of Asia. The knot was later described in Roman historian by Curtius Rufus as comprising of several knots so tightly inter tangled that it was impossible to see how they were fastened. So the ox card stood in the palace of the former kings of Ruia at Godi until the fourth century BC and FI became the province of the Persian Empire. One day a young soldier entered the city and wanted to untie the knot but struggled to do so. So he drew his sword and slashed it open with a single stroke. That soldier was Alexander the great and he did conquer the whole of Asia. Although the prophecy was fulfilled, some people said he actually took out the yoke fixing the car to the knot. But I much prefer the surgical story of using the knife. But going back to mathematics, the first steps towards mathematical theory of knots were taken in 1800 by German mathematician Carl Friedrich Gars and the origins of mo modern knot theory. However, stem from the suggestion by a Scottish mathematician and physicist named William Thompson or Lord Kelvin in 1869. And he posited that Athens might consist of knotted vortex tubes of ether with different elements corresponding to different knots. Hm. I read that and wondered is that a suggestion of string theory. But in response, a contemporary Scottish mathematician and physicist Peter Guthrie Tate made the first systematic attempt to classify not. And although Kelvin's theory was eventually rejected, including ether not theory developed as a pure mathematical discipline for 100s of years, then a major breakthrough came through in New Zealand by Vaughan Jones in 1984 with the introduction of Jones polynomials as new not invariance that led the American mathematical physicist Edward Whitten to discover a connection between knot theory and quantum field theory. Both men were awarded fields of metal in 1990. Another direction, the American mathematician William Thurston made an important link between knot theory and hyperbolic geometry with possible ramifications for cosmology and the stars not theory has actually been applied to biology, to chemistry, mathematics, cosmology, physics.