All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron

TL;DR

The paper demonstrates that all 2-dimensional links in 4-space can be represented within a specific universal 3-dimensional polyhedron, using a novel algebraic encoding method.

Contribution

It introduces a universal 3D polyhedron that contains all 2D links in 4-space and develops a semigroup to classify these links algebraically.

Findings

Any 2D link in 4-space is isotopic to a surface in the universal polyhedron.

A finitely presented semigroup encodes all isotopy classes of 2D links.

The proof uses marked graphs and algebraic structures to represent surfaces in 4-space.

Abstract

The hexabasic book is the cone of the 1-dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3-dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any 2-dimensional link in 4-space is isotopic to a surface in UP. The proof is based on a representation of surfaces in 4-space by marked graphs, links with double intersections in 3-space. We construct a finitely presented semigroup whose central elements uniquely encode all isotopy classes of 2-dimensional links.