  It is not known whether there exists a computable function bounding the
number of Pachner moves needed to connect any two triangulation of a compact
3-manifold. In this paper we find an explicit bound of this kind for all Haken
3-manifolds which contain no fibred submanifolds as strongly simple pieces of
their JSJ-decomposition. The explicit formula for the bound is in terms of the
number of tetrahedra in the two triangulations. This implies a conceptually
trivial algorithm for recognising any non-fibred knot complement among all
3-manifolds.
