I will begin with an introduction to the subfactor classification program, which has two main focuses: restricting the list of possible principal graphs, and constructing examples when the graphs survive known obstructions. I will discuss recent joint work with Liu and Morrison which classifies 1-supertransitive subfactors without intermediates with index in $(3+\sqrt{5},6.2)$. We show there are exactly 3 examples corresponding to a BMW algebra and two "twisted" variations.