Knotting and weak knotting in confined, open random walks using virtual knots

Abstract

We probe the character of knotting in open, confined polymers, assigning knot types to open curves by identifying their projections as virtual knots. In this sense, virtual knots are transitional, lying in between classical knot types, which are useful to classify the ambiguous nature of knotting in open curves.Modelling confined polymers using both lattice walks and ideal chains, we find an ensemble of random, tangled open curves whose knotting is not dominated by any single knot type, a behaviour we call weakly knotted. We compare cubically confined lattice walks and spherically confined ideal chains, finding the weak knotting probability in both families is quite similar and growing with length, despite the overall knotting probability being quite different. In contrast, the probability of weak knotting in unconfined walks is small at all lengths investigated. For spherically confined ideal chains, weak knotting is strongly correlated with the degree of confinement but is almost entirely independent of length. For ideal chains confined to tubes and slits, weak knotting is correlated with an adjusted degree of confinement, again with length having negligible effect.

Revisiting the second Vassiliev (In)variant for polymer knots

“…While these methods yield a definite topological classification (for simple knots that do not share an Alexander polynomial), open curves do not necessarily have a definite knot topology. More rigorously, the ends of chain may be projected to many points on a virtual sphere surrounding the knot, computing a distribution of possible topologies consistent with the curve [15,16] (figure 1(a)). First discussed in the context of protein knotting [17], this is sometimes known as stochastic closure, although the points need not be chosen randomly.…”

“…The second is the case of spherically confined polymers, in which the motion of the ends of the chain through the polymer-filled sphere leads to a diffusive equilibrium of knotting and unknotting. Alexander et al [15] define 'strong' and 'weak' knotting based on the stochastic closure of knots to the surface of a bounding sphere, defining strong knotting when more than half the projections are represented by a single knot type, weak knotting when there is only a plurality, and unknottedness when over half the projections yield the unknot. An unconfined polymer knot is typically strongly knotted when the chain ends are far from the knotted core, but can become weakly knotted as the ends of the chain reptate through the knot as it unties.…”