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Generalized knot complements and some aspherical ribbon disc complements. (English) Zbl 1053.57005
A labeled oriented graph (LOG) is an oriented graph on vertices \(\{1,\dots, n\}\), say, where each oriented edge is labeled by a vertex. Associated with it comes a presentation on generators \(x_1,\dots, x_n\) in one-to-one correspondence with the vertices. For any edge with initial vertex \(i\), terminal vertex \(j\) and label \(k\) we add a relation \(x_i x_k= x_k x_j\). Such a presentation is called a LOG-presentation and the standard 2-complex associated with it is called a LOG-complex. A LOT is a LOG where the underlying graph is a tree. LOT-presentations play a central role in view of the Whitehead conjecture: Any subcomplex of an aspherical 2-complex is itself aspherical. The authors generalize some aspects of standard knot theory to all ribbon-disc complements. They study asphericity of the complement of properly embedded links in certain contractible singular 3-manifolds that should be thought of as replacements of the 3-ball in the classical setting. Then they apply the obtained results to show the asphericity of 2-complexes modelled on LOG’ s that correspond to alternating prime projections on some surface. More precisely, they prove the following result: If \(L\) is an alternating prime arc-projection on a punctured orientable surface, then the LOT-complex associated with \(L\) is aspherical.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M20 Two-dimensional complexes (manifolds) (MSC2010)
Full Text: DOI
[1] DOI: 10.1017/CBO9780511629358.012 · doi:10.1017/CBO9780511629358.012
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