Crossing detection using projections.

<p>(a) A 3 link chain with its vertical projection. A crossing in the projection is shown with a green circle. The crossing in the projection occurs at points and , where the chain is parametrized uniformly from 0 to 3. Since link 1 is under link 3 at the point of projection crossing, 0.29 will appear with a negative sign in the corresponding (eqn 1). (b) The blue chain and the red chain have the exact same vertical projection, however their corresponding matrices are different in sign, as given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0053642#pone.0053642.e224" target="_blank">Eq. 2</a>. This indicates that the over-under sense has changed for the links whose projections are crossing. This in turn indicates that a true crossing has occurred when going from the red conformation to the blue conformation, as opposed to a series of conformations where the chain has navigated to conformations having the opposite crossing sense without passing through itself.