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.

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ABSTRACT. The method of presentation for n-knots is used to classify simple (2q -l)-knots, q > 3, in terms of the Blanchfield duality pairing. As a corollary, we characterize the homology modules and pairings which can arise from classical knots. 0. Introduction. In this paper we use the results and techniques of [4] to give a classification of simple knots in terms of the Blanchfield duality pairing.We work in the piecewise linear category throughout, and it is to be understood that all embeddings, submanifolds, and isotopies are locally flat. An n-knot is an oriented pair (Sn+2, S"), where Sr denotes the r-sphere; two n-knots are equivalent if there is an isomorphism of pairs between them which preserves orientations.A knot of S2q~x in S2q+X is simple if its complement has the homotopy (q -l)-type of a circle; in the terminology of [4], it is a (q -l)-simple (2q -1)-knot. The term simple is due to Levine [8].Let K denote the complement of a simple (2q -l)-knot; if 5 is the integral group ring of the infinite cyclic group, and 50 its field of fractions, then Blanchfield [1] shows that there is a nonsingular pairing of 5-modules V:Hq(K,dK)xHq(K)^R0/R, where K denotes the infinite cyclic cover of K.By Corollary 10.1 of [4], if q > 3, a simple (2q -l)-knot has a presentation with one 0-handle, some (q -1)-and <7-handles, and one (2q -l)-handle. Such a presentation is called simple, and we use it to obtain a matrix presentation of Hq(K) and V; we remark that Hq(K, dK) = Hq(K) by the long exact sequence of homology.By means of the matrix presentation, we show that for q > 3 a simple (2q -l)-knot is determined by its homology module Hq(K) and the duality "

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