  We introduce a method for calculating the extent to which chain non-crossing
is important in the most efficient, optimal trajectories or pathways for a
protein to fold. This involves recording all unphysical crossing events of a
ghost chain, and calculating the minimal uncrossing cost that would have been
required to avoid such events. A depth-first tree search algorithm is applied
to find minimal transformations to fold $\alpha$, $\beta$, $\alpha/\beta$, and
knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a
small fraction of the total distance travelled by a ghost chain. Different
structural classes may be distinguished by the amount of extra uncrossing
distance, and the effectiveness of such discrimination is compared with other
order parameters. It was seen that non-crossing distance over chain length
provided the best discrimination between structural and kinetic classes. The
scaling of non-crossing distance with chain length implies an inevitable
crossover to entanglement-dominated folding mechanisms for sufficiently long
chains. We further quantify the minimal folding pathways by collecting the
sequence of uncrossing moves, which generally involve leg, loop, and elbow-like
uncrossing moves, and rendering the collection of these moves over the unfolded
ensemble as a multiple-transformation "alignment". The consensus minimal
pathway is constructed and shown schematically for representative cases of an
$\alpha$, $\beta$, and knotted protein. An overlap parameter is defined between
pathways; we find that $\alpha$ proteins have minimal overlap indicating
diverse folding pathways, knotted proteins are highly constrained to follow a
dominant pathway, and $\beta$ proteins are somewhere in between. Thus we have
shown how topological chain constraints can induce dominant pathway mechanisms
in protein folding.
