Symmetric Surfaces in S^4

Date

2025

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.

Description

Keywords

Knot Theory, Low Dimensional Topology

Graduation Month

May

Degree

Doctor of Philosophy

Department

Department of Mathematics

Major Professor

David R. Auckly

Date

Type

Dissertation

Citation