Tantalizing Patterns Created by Curves on Surfaces

Consider an orientable surface with negative Euler characteristic, a minimal set of generators of its fundamental group, and a hyperbolic (i.e., constant negative curvature) metric. Each free homotopy class of closed oriented curves on it determines three numbers:

-the minimal geometric self-intersection number,
-the geometric length (computed by the hyperbolic metric), and
-the word length (i.e., the minimum number of letters needed to express the class as a cyclically reduced word in the generators and their inverses).

We will explain how we computed or approximated these three numbers and how these computations led to counterexamples to existing conjectures, to the formulation of new conjectures, and, at times, to new theorems.

(These results are joint work with various collaborators; mainly  Steven Lalley, Anthony Phillips and Rachel Zhang)

Bio
https://www.math.stonybrook.edu/~moira/