It is a well-known result by Edmonds that every periodic knot of genus g bounds 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.

The author was partially supported by the Simons Foundation, Award ID 524394, and by the NSF, Grant No. DMS-1906413.