A few notes about polymer knots

Basic theoretical results on knots in a single polymer chain are reviewed. The probability of a knot upon random chain closure and the effect of a preexisting knot on the physical characteristics of a molecule are considered. The probability of a knot in the linear chain is related to knot entropy and the minimum work required for a knot to become untied in a thermodynamically equilibrium environment. However, when chain topology is fixed by end closure, it controls, for example, the average radius of gyration of a coil: a coil appears to be more compact as compared with a phantom one when a knot is more complex than a certain threshold, and more swollen in the opposite case.