Drawing curves on surfaces or: how I learned to stop worrying and love train-tracks

Illustration is a vital part of outreach, pedagogy, exposition, and experimentation in mathematics. But illustration also plays a key role in the informal communication between, and within, mathematicians. In low-dimensional topology, certain cartoons of surfaces are used so often that they are essentially part of the Ph.D. syllabus. I'll give an example of how I combined approximately 100 doodles to give a delicate proof by cases, how I fought to remove them, how I failed, and why the doodles did not appear in the final paper.