Aspherical labelled oriented trees and knots.

*(English)*Zbl 0983.57003An 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.

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.

Reviewer: Riccardo Piergallini (Camerino)