Universal quantum knot invariants
In mathematics, a knot is what results from taking a piece of rope, moving it in a certain way, intertwining it as much as desired, and finally joining the ends of the rope. Two knots are considered equal if it is possible, without untying the ends, to reshape the first one by moving the rope and obtain a knot that has the same form as the second one.
Let us suppose that, given another pair of knots, after hours and hours patiently moving one of the knots in an attempt to obtain the other, we still do not succeed. One might start thinking that it is not possible. But how can we be certain that it is truly impossible, and that one extra hour of manipulating ropes will not eventually provide an affirmative answer to the question?
To address this issue, a significant part of the mathematical theory of knots involves creating “machines” (that is, procedures, recipes, algorithms, etc.) where for each of them the input is a certain knot, and the output is a certain quantity (for example, a number) associated with it, with the property that “equal” knots have the same output. If the quantities associated to two knots are different, we can then conclude that the knots are not the same.
Jorge Becerra Garrido focussed on such invariants that are constructed using algebraic structures that arose in the 1980s inspired by ideas from theoretical physics.