Jeffrey Meier

A foundational theorem in knot theory states that every knot in 3-space can be expressed uniquely as a connected sum of non-trivial prime knots. To this day, very little is known about the extent to which a similar result might hold for embeddings of closed surfaces in 4-space. For example, it is not known whether the unknotted 2-sphere admits a non-trivial connected sum decomposition, and it is conjectured that every knotted projective plane is the connected sum of a knotted 2-sphere and an unknotted projective plane. In this talk, I’ll survey that pathologies that arise concerning notions of ‘decomposability’ or ‘irreducibility’ for surface-knots in 4-space, and I’ll present an infinite family of knotted Klein bottles that are indecomposable and have order-4 meridians.