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Aspherical labelled oriented trees and knots. (English) Zbl 0983.57003
An oriented graph $$\Gamma$$ with vertices $$\{x_1,\dots,x_n\}$$ is called a labelled oriented graph (LOG) if a $$\{1, \dots, n\}$$-labelling is given on its edges. In such a case, $$P_\Gamma$$ is the group presentation with generators $$\{x_1, \dots, x_n\}$$ and a relation $$x_i = x_jx_kx_j^{-1}$$ for each edge from $$x_i$$ to $$x_k$$ with label $$j$$. Moreover, $$K_\Gamma$$ denotes the standard 2-complex associated to $$P_\Gamma$$.
The $$K_\Gamma$$’s corresponding to labelled oriented trees (LOT) arise in a natural way as spines of complements of all ribbon-disks in $$B^4$$ [J. Howie, Topology 22, 475-485 (1983; Zbl 0524.57002)]. Then, their asphericity coincides with the ribbon-disk conjecture and is related to the Whitehead asphericity conjecture [J. Howie, Trans. Am. Math. Soc. 289, 281-302 (1985; Zbl 0572.57001)].
A LOT is said to be: reducible, there is a vertex of valence one $$x_i$$ such that $$i$$ does not occur as a label of any edge; compressed, if no edge is a loop or is labelled with the index of one of its end-points; injective, if the labelling map is injective.
A 2-complex $$K$$ is called: diagrammatically reducible (DR), if any spherical diagram over $$K$$ is reducible by a folding operation; diagrammatically aspherical (DA), if any spherical diagram over $$K$$ can be converted into a reducible one by diamond moves. Of course, DR $$\Rightarrow$$ DA $$\Rightarrow$$ aspherical.
The main result of the paper under review is the following: if $$\Gamma$$ is a compressed injective labelled oriented forest which does not contain any reducible LOT, then the 2-complex $$K_\Gamma$$ is DR.
As a corollary the author obtains a new combinatorial proof of the asphericity of complements of alternating knots [C. M. Weinbaum, Proc. Am. Math. Soc. 30, 22-26 (1971; Zbl 0228.55004)], based on DA of Wirtinger spines.

##### MSC:
 57M20 Two-dimensional complexes (manifolds) (MSC2010) 20F06 Cancellation theory of groups; application of van Kampen diagrams 20F05 Generators, relations, and presentations of groups 20F65 Geometric group theory
##### Keywords:
2-complexes; asphericity; knot groups
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