Aspherical Labelled Oriented Trees and Knots

Abstract

The question of whether ribbon-disc complements-or, equivalently, standard 2-complexes over labelled oriented trees-are aspherical is of great importance for Whitehead's asphericity conjecture and, if solved affirmatively, would imply a combinatorial proof of the asphericity of knot complements. We present here two classes of diagrammatically reducible labelled oriented trees.

Injective labeled oriented trees are aspherical

“…The next result is proved in Section 3 of [9]. The situation is more complicated in the presence of a boundary reducible sub-LOT.…”

“…The material in this section is of independent interest with possible applications not directly connected with the study of labeled oriented trees. Section 4 introduces altered LOTpresentations and contains Theorem 4.1, a result shown by Huck and the second author [9]. There it was used to show that prime injective labeled oriented trees are aspherical (prime means that the labeled oriented tree does not contain sub-LOTs).…”

Generalized Knot Complements and Some Aspherical Ribbon Disc Complements

“…On the other hand the LOT-complex of Fig. 6 is DR, which is a consequence of Theorem 1.1 of [9]. In fact the LOT-complex is injective and does not contain a reducible (as explained in [9]) sub-LOT.…”

“…The authors know of no other way to show the asphericity of the corresponding LOT, the methods in [9] fail in this case.…”

Spherical diagrams and labelled oriented trees

“…The conjecture that all LOT complexes are aspherical is known in knot theory as the ribbon-disc conjecture. It is also an important subcase of the Whitehead conjecture, that is, the conjecture that all subcomplexes of aspherical 2-complexes are aspherical [3,4,6,7].…”