On the minimum dilatation of braids on punctured discs

TL;DR

This paper determines the smallest dilatation values for pseudo-Anosov braids on punctured discs with 3 to 8 punctures, using an elementary proof based on the Lefschetz formula.

Contribution

It extends known results to new cases of punctured discs and provides an elementary proof method for calculating minimal dilatations.

Findings

Minimum dilatations for n=3,6,7,8 punctured discs identified

Results include previous findings for n=4 and 5

Proof employs Lefschetz formula, simplifying previous approaches

Abstract

We find the minimum dilatation of pseudo-Anosov braids on n-punctured discs for 3 <= n <= 8. This covers the results of Song-Ko-Los (n=4) and Ham-Song (n=5). The proof is elementary, and uses the Lefschetz formula.