Smoothing curves carefully

TL;DR

This paper proves that carefully smoothing intersection points on closed curves reduces self-intersections by one, confirming that shortest geodesics with at least k self-intersections have exactly k, for hyperbolic and Riemannian surfaces.

Contribution

It establishes an elementary topological fact and applies it to confirm a longstanding conjecture about the self-intersection number of shortest geodesics.

Findings

Smoothing reduces self-intersections by exactly one.

Shortest geodesics with at least k self-intersections have exactly k.

The result applies to hyperbolic and arbitrary Riemannian metrics.

Abstract

This paper proves an elementary topological fact about closed curves on surfaces, namely that by carefully smoothing an intersection point, one can reduce self-intersection by exactly $1$. This immediately implies a positive answer to a problem first raised by Basmajian in the 1990s: among all closed geodesics of a hyperbolic surface that self-intersect at least $k$ times, does the shortest one self-intersect exactly $k$ times? The answer is also shown to be positive for arbitrary Riemannian metrics.