DNA knots can arise in vivo and in vitro in various biological reactions involving circular DNA molecules. So for example site specific recombination enzymes are known to produce specific families of knots. By identification of the formed knot types it is possible to conclude about the mechanism of action of a given enzyme and about the overall shape of supercoiled circular DNA molecules at the moment of knotting. Recently DNA knots arising in vivo during the process of DNA replication have been characterised. These knots are localised within so called replication bubbles and show predominantly positive crossings. Analysis of formed knot types allows us to conclude about the structural organisation of replicating circular DNA molecules in living cells.

Ideal knots are defined as trajectories of shortest tubes with a constant diameter which can be still closed into a given type of knot. Ideal knots have a number of intriguing mathematical and physical features, including a direct correspondence with the time-averaged shapes of random knots. Recently I noticed that writhe of ideal knots is quantized into units of 4/7 and 10/7 whereby the total writhe of a knot is determined by a simple arithmetic sum of four elementary types of crossings which need to be "removed" to convert a standard minimal crossing diagram of a given knot into a diagram of a trivial knot without nugatory crossings. There are parallel and antiparallel crossings of positive and negative sign. Interestingly parallel and antiparallel crossings of the same sign show frequently superposition of states.

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