Symmetric Surfaces in S^4
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Classical knots and their invariants provide key insights into many questions in lowdimensional topology. One invariant of knots crucial to the study of surfaces in 4-manifolds is 4-genus. We combine two variants of 4-genus (equivariant 4-genus and double-slice genus) to create a new variant of symmetric knots, which we call equivariant double-slice genus. This dissertation works to initiate the study of this new knot invariant. Namely, we introduce the new invariant, prove elementary results about it, and prove a useful lower bound for it. The lower bound we construct for the equivariant double-slice genus is easily computable and powerful enough to effectively distinguish the equivariant double-slice genus from existing knot invariants. Additionally, using equivariant double-slice genus as the primary obstructive invariant, we begin to study equivariant embeddings of closed surfaces in the 4-sphere. Specifically, we prove the existence of equivariantly embedded 2-spheres in the 4-sphere which are isotopic but not equivariantly isotopic and remain equivariantly distinct after many internal stabilizations.