On computational complexity of plane curve invariants

Abstract

The theory of generic smooth closed plane curves initiated by Vladimir Arnold is a beautiful fusion of topology, combinatorics, and analysis. The theory remains fairly undeveloped. We review existing methods to describe generic smooth closed plane curves combinatorially, introduce a new one, and give an algorithm for efficient computation of Arnold's invariants. Our results provide a good source of future research projects that involve computer experiments with plane curves. The reader is not required to have background in topology and even undergraduate students with basic knowledge of differential geometry and graph theory will easily understand our paper.