The Local Structure of Injective LOT-Complexes

Labeled oriented trees, LOT's, encode spines of ribbon discs in the 4-ball and ribbon 2-knots in the 4-sphere. The unresolved asphericity question for these spines is a major test case for Whitehead's asphericity conjecture. In this paper we give a complete description of the link of a reduced injective LOT complex. An important case is the following: If $\Gamma$ is a reduced injective LOT that does not contain boundary reduced sub-LOTs, then $lk(K(\Gamma))$ is a bi-forest. As a consequence $K(\Gamma)$ is aspherical, in fact DR, and its fundamental group is locally indicable. We also show that a general injective LOT complex is aspherical. Some of our results have already appeared in print over the last two decades and are collected here.