Department of
Applied
Mathematics

Spirals

Eric Sedgwick

Abstract

This is joint work with Jan Kyncl, Marcus Schaefer, Daniel Stefankovic.

In a surface, draw a pair of curves with a large number of intersections. Must the drawing contain any particular sub--structures? A spiral is especially desirable because it can be used to simplify drawings in a number of contexts (string graphs, quadratic word equations).

However, we show that for every n there is a pair of curves in the plane with n intersections, but that do not form a spiral. This proof utilizes train tracks, a topological tool that provides a compressed representation for densely packed curves.