  It was recently shown that there exists an explicit bound for the number of
Pachner moves needed to connect any two triangulation of any Haken 3-manifold
which contains no fibred sub-manifolds as strongly simple pieces of its
JSJ-decomposition. In this paper we prove a generalisation of that result to
all knot complements. The explicit formula for the bound is in terms of the
numbers of tetrahedra in the two triangulations. This gives a conceptually
trivial algorithm for recognising any knot complement among all 3-manifolds.
