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On the Non-Orientable Equivariant 4-Genus of a Periodic Knot
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We prove the existence of an infinite family of periodic knots, iterated by integers greater than one, for which the non-orientable 4-genus and the equivariant non-orientable 4-genus actually differ. This work stands in contrast to the result by Edmonds which showed that the Seifert genus and equivariant Seifert genus of a periodic knot must always agree. Furthermore, this result is analogous to previous results pertaining to the smooth 4-genus (and its equivariant version) and the non-orientable 3-genus (and its equivariant version). The proof of our result hinges upon finding obstructions to the existence of particular types of lattice embeddings, which arise by assuming, towards contradiction, that the non-orientable 4-genus and the equivariant non-orientable 4-genus of a particular periodic knot agree.