Periodic spanning surfaces of periodic knots

TL;DR

This paper explores the limitations of extending Edmonds' theorem from orientable to nonorientable spanning surfaces of periodic knots, showing that the genus equivalence does not hold in the nonorientable case.

Contribution

It demonstrates that the difference in the first Betti number between equivariant and non-equivariant nonorientable spanning surfaces can be arbitrarily large, highlighting a fundamental distinction.

Findings

The genus equivalence does not extend to nonorientable surfaces.

The Betti number difference can be arbitrarily large.

Provides explicit examples illustrating this disparity.

Abstract

Edmonds famously proved that every periodic knot of genus g possesses an equivariant Seifert surface of genus g. We show that this is not true if one instead considers nonorientable spanning surfaces of a periodic knot. We demonstrate by example that the difference between the first Betti number of an equivariant and a nonequivariant nonorientable spanning surface of a periodic knot, can be arbitrarily large.