Abstract: It may seem unlikely that rich mathematical structures still remain to be uncovered in small arrays of classical, linear oscillators. However, oscillators that combine non-reciprocal elements with loss (or gain) have provided a number of surprises in recent years. My talk will describe how these systems are naturally connected to braid groups, knot theory, and other topological structures. I will give a pedagogical introduction to this topic (with lots of pictures!) and will present measurements of these phenomena in a cavity optomechanical system.