Flying circles and non-commutative algebra

I will describe a machine that lets us go between certain problems in topology and problems in non-commutative algebra. One main question of interest to topologists is how to distinguish between different knotted objects – a powerful tool that has been developed to address this are the so-called universal finite-type invariants. On the algebraic side, this structure translates to finding solutions of sets of equations in quantum algebra, in the graded setting. In this talk, we will in particular consider a certain class of knotted tubes in 4-space, pictured as movies of flying circles, and show how they are related to a class of equations originally introduced by Masaki Kashiwara and Michele Vergne.