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.