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 mini...

Using the theory of Properly Embedded Graphs developed in an earlier work we define involutory duality on set labeled non-crossing trees that lifts obvious unlabeled trees. The is a free ternary magma with one generator and this instance defined any such magma. Any two magmas are isomorphic via unique isomorphism call structural bijection. Besides also consider as trees, quadrangular dissection...