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

Kearton, C.; Kurlin, V.

doi:10.2140/agt.2008.8.1223

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

Kearton, C.; Kurlin, V.

Authors

C. Kearton

V. Kurlin

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.

Citation

Kearton, C., & Kurlin, V. (2008). All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron. Algebraic & geometric topology, 8(3), 1223-1247. https://doi.org/10.2140/agt.2008.8.1223

Journal Article Type	Article
Publication Date	Jul 1, 2008
Deposit Date	Dec 7, 2010
Publicly Available Date	Apr 5, 2013
Journal	Algebraic and Geometric Topology
Print ISSN	1472-2747
Electronic ISSN	1472-2739
Publisher	Mathematical Sciences Publishers (MSP)
Peer Reviewed	Peer Reviewed
Volume	8
Issue	3
Pages	1223-1247
DOI	https://doi.org/10.2140/agt.2008.8.1223
Keywords	2-knot, 2-link, Handle decomposition, Hexabasic book, Marked graph, Singular link, Uuniversal polyhedron, 3-page book, 3-page embedding, Universal semigroup.