×

zbMATH — the first resource for mathematics

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.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M20 Two-dimensional complexes (manifolds) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1017/CBO9780511629358.012 · doi:10.1017/CBO9780511629358.012
[2] DOI: 10.1007/978-3-662-12494-9 · doi:10.1007/978-3-662-12494-9
[3] DOI: 10.1007/BF01456216 · Zbl 0477.20019 · doi:10.1007/BF01456216
[4] DOI: 10.1007/978-1-4613-9586-7_2 · doi:10.1007/978-1-4613-9586-7_2
[5] DOI: 10.1017/CBO9780511629358.003 · doi:10.1017/CBO9780511629358.003
[6] DOI: 10.1016/0040-9383(83)90038-1 · Zbl 0524.57002 · doi:10.1016/0040-9383(83)90038-1
[7] DOI: 10.1090/S0002-9947-1985-0779064-8 · doi:10.1090/S0002-9947-1985-0779064-8
[8] G. Huck and S. Rosebrock, Combinatorial and Geometric Group Theory, London Math. Soc. Lecture Note Ser. 204, eds. J. Howie, A. Duncan and N. Gilbert (Cambridge University Press, London, 1995) pp. 174–183.
[9] DOI: 10.1017/S0013091599000474 · Zbl 0983.57003 · doi:10.1017/S0013091599000474
[10] Rolfsen D., Knots and Links (1976) · Zbl 0339.55004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.