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

Kearton, Cherry; Kurlin, Vitaliy

doi:10.2140/agt.2008.8.1223

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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

DOI: 10.2140/agt.2008.8.1223

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

References

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/

Volume 8, issue 3 (2008)