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The student of the University of Texas at Austin, is Lisa Piccirillo , ran into one of those enigmas that populate the mathematics , the so-called knot of Conway . Its intention, as a mere pastime, was to apply what you have learned during your phd, so he put hands to the work. In a week, he had the solution. When he showed it to his professor, he could not believe that Piccirillo had managed: to resolve a enigma mathematician that had for decades been open. Their findings have been published in march in the journal “Annals of Mathematics”.
To understand the achievement of the promising student, one must know a few basic terms on the theory of knots . In mathematics, a knot would be a rope tied in the have stuck the ends together. That is to say, has neither beginning nor end. On that, the theory of knots examines the transformations that can be made to that string bending it, stretching it it, twisting it… but always without actually cutting it. From there, the premise is you try to test is if, given knots , it is possible to get one of them from the deformations of the other . If that can be done, then we are between two knots to be equivalent.
In mathematics we use the so-called ” invariants of knots “, which are functions that assign a value to each node, thus transforming the string into mathematical formulas. But as if from a knot you can give another, if a given invariant assigns different values to two nodes, then it is not possible to deform a knot in the other . That is to say, are not equivalent.
in Addition, the knots may have (or not) a series of properties. Among them, the ability to be “slice” or not . To understand this concept it is necessary to imagine the knot in a space of four dimensions: a knot to be slice if it is the edge of a disk in this space . And this is not easy to imagine, hence its complexity. But here the invariants come to our aid. So far, mathematicians have determined that 2.977 of 2.798 knots with less than 13 crossings have the property of being a slice or not. But there that the of Conway (devised by the mathematician John Horton Conway , recently died of coronavirus), 11 crossings, it had been impossible to determine in 50 years.
Example of a knot of Conway – Wikicommons
Piccirillo applied a original approach , in addition to draws attention due to its simplicity. Explained in a simple way, based on the fact that the knots trace equivalent have that to have always the condition of which, or both, are slice, or both not, he used another knot equivalent to verify that, indeed, the knot of Conway is not a slice! . Something that will have an effect, as it is one of the foundations of the study of mutation in the theory of knots. Although she was a mere distraction.