Polymer molecules in dilute solution are long, flexible objects which can be highly self-entangled. If a ring closure reaction occurs (turning the linear polymer into a ring polymer) then the entanglement can be captured as a knot. Knots have been observed experimentally in circular DNA and their frequency of occurence has been studied as a function of the molecular weight of the DNA and the ionic strength of the solution. For an interesting account of how the topology of closed circular DNA can be used to probe the action of enzymes such as the topoisomerases, see a review by De Witt Sumners in Notices of the AMS, May 1995.

In the 1960s Max Delbruck, Ed Wasserman and Harry Frisch suggested that the knot probability in ring polymers would approach unity as the length of the ring approached infinity. This question was studied numerically (mainly using Monte Carlo methods), and was finally settled in 1988 when it was shown that for a simple closed curve embedded in Z3 the knot probability goes to unity exponentially rapidly as the length of the curve approaches infinity. There is a nice review of this area, at a not-too-technical level, by Brian Hayes in American Scientist, Nov-Dec 1997.

Just as a single ring polymer can be knotted, so two or more ring polymers can be linked. Again, linked pairs of circles have been seen in experiments on circular DNA molecules. For some beautiful graphics of links see the work of Rob Scharein.