Minimal Penner dilatations on nonorientable surfaces

TL;DR

This paper determines the minimal dilatations of pseudo-Anosov mapping classes from Penner's construction on nonorientable surfaces, revealing two accumulation points and employing Alexander polynomial techniques.

Contribution

It identifies the minimal Penner dilatations on nonorientable surfaces and shows they have two accumulation points, unlike the orientable case.

Findings

Minimal Penner dilatations have exactly two accumulation points.

Pseudo-Anosov dilatations are represented as roots of Alexander polynomials.

Comparison of dilatations uses skein relations for Alexander polynomials.

Abstract

For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner's construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points, in contrast to the case of orientable surfaces where there is only one accumulation point. One of our key techniques is representing pseudo-Anosov dilatations as roots of Alexander polynomials of fibred links and comparing dilatations using the skein relation for Alexander polynomials.