Geodesic
All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron
Kearton, Cherry1 ; Kurlin, Vitaliy1
1Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom
Algebraic and Geometric Topology, Tome 8 (2008) no. 3, pp. 1223-1247
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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.

DOI : 10.2140/agt.2008.8.1223
Keywords: 2-knot, 2-link, handle decomposition, hexabasic book, marked graph, singular link, universal polyhedron, 3-page book, 3-page embedding, universal semigroup

Kearton, Cherry  1   ; Kurlin, Vitaliy  1

1 Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom 
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Kearton, Cherry; Kurlin, Vitaliy. All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron. Algebraic and Geometric Topology, Tome 8 (2008) no. 3, pp. 1223-1247. doi: 10.2140/agt.2008.8.1223

[1] V I Arnol’D, A N Varchenko, S M Guseĭn-Zade, Osobennosti differentsiruemykh otobrazhenii [Singularities of differentiable mappings], Nauka (1982) 304

[2] I A Dynnikov, Three-page approach to knot theory. Coding and local motions, Funktsional. Anal. i Prilozhen. 33 (1999) 25, 96

[3] I A Dynnikov, A three-page approach to knot theory. The universal semigroup, Funktsional. Anal. i Prilozhen. 34 (2000) 29, 96

[4] T. Fiedler, V. Kurlin, Fiber quadrisecants in knot isotopies, to appear in J. Knot Theory Ramifications

[5] T. Fiedler, V. Kurlin, A one-parameter approach to links in solid torus

[6] R H Fox, J W Milnor, Singularities of $2$–spheres in $4$–space and cobordism of knots, Osaka J. Math. 3 (1966) 257

[7] L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697

[8] C Kearton, W B R Lickorish, Piecewise linear critical levels and collapsing, Trans. Amer. Math. Soc. 170 (1972) 415

[9] V Kurlin, Three-page encoding and complexity theory for spatial graphs, J. Knot Theory Ramifications 16 (2007) 59

[10] F J Swenton, On a calculus for $2$–knots and surfaces in $4$–space, J. Knot Theory Ramifications 10 (2001) 1133

[11] V V Vershinin, V A Kurlin, Three-page embeddings of singular knots, Funktsional. Anal. i Prilozhen. 38 (2004) 16, 95

[12] K Yoshikawa, An enumeration of surfaces in four-space, Osaka J. Math. 31 (1994) 497

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