Symmetric Surfaces in S^4

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.