#### Volume 8, issue 3 (2008)

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

### Cherry Kearton and Vitaliy Kurlin

Algebraic & Geometric Topology 8 (2008) 1223–1247
##### 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 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.

##### Keywords
2-knot, 2-link, handle decomposition, hexabasic book, marked graph, singular link, universal polyhedron, 3-page book, 3-page embedding, universal semigroup
##### Mathematical Subject Classification 2000
Primary: 57Q45, 57Q35, 57Q37
##### Publication
Received: 7 April 2008
Revised: 7 June 2008
Accepted: 13 June 2008
Published: 26 July 2008
##### Authors
 Cherry Kearton Department of Mathematical Sciences, Durham University Durham DH1 3LE United Kingdom http://www.maths.dur.ac.uk/~dma0ck/ Vitaliy Kurlin Department of Mathematical Sciences, Durham University Durham DH1 3LE United Kingdom http://maths.dur.ac.uk/~dma0vk/