It may seem unlikely that rich mathematical structures remain to be uncovered in classical harmonic oscillators. Nevertheless, oscillators that combine non-reciprocity and loss have provided a number of surprises in recent years. I will describe a collaborative effort between theory and experiment that has led to a concise understanding of why these systems naturally exhibit braids, knots, and other topological structures. I will also present measurements of these structures (using a cavity optomechanical system), and will describe their potential application in various settings. This will be a pedagogical introduction to all of these topics, with lots of pictures and videos!