22/06/2020

Young man solves problem that has gone unanswered for half a century.

Foto: Ian MacLellan/ Quanta Magazine

“I thought of it as homework,” young Lisa Piccirillo told BBC News , about solving a mathematical problem that had remained unanswered for half a century in less than a week. After learning about the enigmatic 11-crossing knot proposed by British mathematician John Conway, who died in April of this year from Covid-19 , at a seminar, the then doctoral student at the University of Texas used her free time to answer: is the “Conway knot” slicable?

To study the problem, it is necessary to understand topology, the branch of mathematics to which it belongs. This area studies the properties that persist after the continuous deformation of geometric objects, twisting or stretching them, but without breaking them. Within topology is knot theory, where the object of study, the knot, is very similar to those in real life.

What deformations can we produce in a rope by bending, stretching, or compressing it? This is the basis of the main questions in knot theory. In 1970, Conway presented a knot with 11 crossings, and since then, mathematicians have tried, unsuccessfully, to discover whether or not it was possible to slice it. However, its "slicable" property has nothing to do with the possibility of cutting the knot in half, but rather with "slices" distributed across the four dimensions of the world – in topology, time is considered this fourth part of the universe.

Among 2,978 nodes with fewer than 13 crossings, the Conway node was the only one whose "slicable" status was unknown, explained mathematician Marithania Silvero of the Institute of Mathematics at the University of Seville, Spain.

To answer the golden question, Lisa Piccirillo developed an ingenious method. She replaced Conway's knot with another one she invented, in which the slice property was easier to study. When presenting the result of her simple "homework," which showed that the British mathematician's knot was not "slicable," Lisa received an enthusiastic reaction from Professor Cameron Gordon of the University of Texas mathematics department, who began shouting, "Why aren't you more excited? This has to be published in the Annals! " (referring to the journal Annals of Mathematics ).

In August 2018, the solution that completed the classification of knots with fewer than 13 slittable or non-slicable crossings was submitted to the specialized journal and, after the entire review process, was published in February 2020 in the Annals of Mathematics (as the professor predicted). This achievement also boosted Lisa's career, as she received a job offer from the Massachusetts Institute of Technology (MIT) just 14 months after completing her doctorate.

Source: BBC News and Quanta Magazine