This question may sound silly. How does it come that the DNA molecules are not totally entangled with time in the cell ? If I take long strings in a box and shake it, that would create knots, and I wouldn't be able to easily isolate each string apart.
But during mitoses, the chromosomes are totally isolated from each others while otherwise, the DNA is freely floating in the nucleus.
I think a key factor is that DNA molecules are not passive bits of string that are left to move freely (which is what causes knots in string-like things that are left alone too long).
For one thing the DNA molecule is an active part of the cell metabolism, as it is constantly being transcribed (to generate RNA and proteins) or copied or corrected and so on. And as part of its role in metabolism the DNA molecule is closely associated with RNA molecules, proteins and specifically proteins that the DNA is wrapped around called "histones". The Wikipedia page for "Chromatin" (what chromosomes are made of, basically) describes various aspects of this organization, including:
That DNA which codes genes that are actively transcribed ("turned on") is more loosely packaged and associated with RNA polymerases (referred to as euchromatin) while that DNA which codes inactive genes ("turned off") is more condensed and associated with structural proteins (heterochromatin).
So DNA is constantly being manipulated by all kinds of proteins and enzymes that affect its shape and position in space; it isn't just lying there. And insofar as it might get into knots, that would likely be part of the manipulations of the enzymes and proteins, or would fairly easily be dealt with by them. This doesn't mean DNA doesn't have knots however, in fact it appears it very much can:
DNA AND KNOT THEORY (from The Institute for Environmental Modelling at the University of Tennessee)
Conclusions: Principles of topology give cell biologists a quantitative, powerful, and invariant way to measure properties of DNA. Principles of knot theory have helped elucidate the mechanisms by which enzymes unpack DNA. Additionally, topological methods have been influential in determining the left handed winding of DNA around histones. Measuring changes in crossing number have also been instrumental in understanding the termination of DNA replication and the role of enzymes in recombination.
(note this paper looks at DNA in E.coli, which does not organize its DNA in chromatin the way Eukaryotes do).
Untangling DNA (from the Nature blog CreatureCast, 2013)
This link includes a video about topoisomerases, enzymes also referred to in the previous link that unentangle DNA.
A Monte Carlo Study of Knots in Long Double-Stranded DNA Chains (PLOS Computational Biology, 2016)
Quote from the abstract:
Even though our coarse-grained model is only based on experimental knotting probabilities of short DNA strands, it reproduces the correct persistence length of DNA. This indicates that knots are not only a fine gauge for structural properties, but a promising tool for the design of polymer models.
The active site of the SET domain is constructed on a knot (Nature Structural Biology, 200)
A knot or not a knot? SETting the record ‘straight’ on proteins (Computational Biology and Chemistry, 2003)
These papers discuss potential knots within a histone protein, not DNA, but illustrates that topology and knots can be important in macromolecules:
A knot within the SET domain helps form the methyltransferase active site, where AdoHcy binds and lysine methylation is likely to occur.
A novel knot found in the SET domain is examined in the light of five recent crystal structures and their descriptions in the literature. Using the algorithm of Taylor it was established that the backbone chain does not form a true knot.
This link is a bit painful to read (and from 1985 so I don't know if it was superseded, but it has over 200 citations) so I will leave it at the title.
And finally this article, which I looked at last but probably should have looked at first because the first sentence of the abstract literally answers your question :
Direct observation of DNA knots using a solid-state nanopore (Nature Nanotechnology, 2016)
Long DNA molecules can self-entangle into knots. Experimental techniques for observing such DNA knots (primarily gel electrophoresis) are limited to bulk methods and circular molecules below 10 kilobase pairs in length. Here, we show that solid-state nanopores can be used to directly observe individual knots in both linear and circular single DNA molecules of arbitrary length.
