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

Abstract

All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron CHERRY KEARTON VITALIY KURLINThe 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 closed 2-dimensional surface 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 surfaces.

A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages

“…Extend our algorithm for drawing graphs in 3 pages to drawing 2-dimensional surfaces in a universal 3D polyhedron from (Kearton and Kurlin, 2008).…”

“…A stronger result in (Kearton and Kurlin, 2008) says that all isotopies between 3-page embeddings are realizable in a 3-dimensional polyhedron (a hexabasic book).…”

Applying Lipson's state models to marked graph diagrams of surface-links

“…[2]). It is known [4,7,8,14] that any surface-link L can be deformed into a surface-link L ′ , called a hyperbolic splitting of L, by an ambient isotopy of R 4 in such a way that the projection p : L ′ → R to the fourth coordinate satisfies the following:…”

“…6. It is known that two admissible marked graph diagrams present equivalent surface-links if and only if they are related by a finite sequence of Yoshikawa moves [8,9,16].…”

“…Local moves on marked graph diagrams introduced in Yoshikawa's paper [17] are so-called Yoshikawa moves. It is known that two marked graph diagrams present equivalent surfacelinks if and only if they are related by a finite sequence of Yoshikawa moves [8,16]. So one can use marked graph diagrams in order to define surfacelink invariants.…”