On intersections of closed curves on surfaces

TL;DR

This paper solves the problem of determining the minimal intersection points of two closed curves on a surface using Nielsen theory, and explores the Wecken property and self-intersection issues.

Contribution

It provides explicit formulas for minimal intersection numbers based on Nielsen and Reidemeister invariants, and shows the Wecken property does not always hold.

Findings

Minimal intersection number expressed via Nielsen and Reidemeister numbers

The Wecken property fails for some pairs of curves

All non-vanishing Nielsen indices are ±1

Abstract

The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is called the Nielsen number. If each Nielsen class consists of a unique point and has a non-vanishing index after a suitable deformation of the pair of curves, one says that {\it the Wecken property holds}. We compute the minimal number of intersection points in terms of the Nielsen numbers and the Reidemeister numbers. In particular, we prove that the Wecken property does not hold for some pairs of closed curves. Moreover, all the non-vanishing indices of the Nielsen classes equal $\pm1$, while the non-vanishing Jezierski semi-indices equal 1. Similar questions are studied for the self-intersection problem of a curve on a surface.