Summer of Math Exposition

Tanglegrams are a way to compare evolutionary histories of species in ecology. Drawings of hosts and parasites are compared to infer the amount of non-ancestral gene transfer between species. The related leaves in two evolutionary trees are connected by edges. The number of crossings between the edges represent the evolutionary "distance" and the amount of horizontal gene transfer needed to explain the coevolution of the hosts and parasites. However, this comparison depends on the drawings of the trees. Another drawing of the trees leads to a different number of crossings. Can we redraw the trees to have the least number of crossings? Will this change the biological conclusion? We introduce the problem as a game. Given a drawing, try to click on the nodes to create a drawing with the smallest number of crossings. After playing the game for a while, the player will be naturally curious about many aspects of the game --- what is the least number of crossings for a particular game, what is the smallest number of clicks to solve the problem etc. We will use this to explain the science and the open questions. 1. The evolutionary biology and why these shapes are of interest in science. 2. The mathematics of the number of drawings, number of crossings etc. 3. The algorithms, the complexity and the many approaches for the solution. This is a large area and there are many angles to the exposition. The submission has only the simple version of the game. There are many more versions of the game, which will be introduced as levels.