On the Non-Orientable Equivariant 4-Genus of a Periodic Knot
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Authors
Grove, Taran
Issue Date
2022
Type
Thesis
Language
Keywords
Equivariant , Genus , Knot Theory , Topology
Alternative Title
Abstract
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.