So I am taking a course in DNA replication and repair. And we are talking about catenanes forming when DNA replicates (two circles of dsDNA interlinked) How is this possible?
The first DNA circle is double-stranded. If you could melt the double-helix completely you would not be able to pull the two stands apart without breaking the sugar-phosphate backbone of at least one of the two strands. This is a topological problem, the two strands are linked to each other.
Now consider DNA replication. In the simplest example there is a single origin of replication and there will be two replication forks proceeding around the circular genome in opposite directions. This is exactly what happens during DNA replication in the bacterium E. coli. The DNA double helix of the parental molecule "melts", the RNA primers are synthesized by primase, and the DNA polymerase complexes initiate synthesis. Behind each replication fork there are now two hybrid DNA strands, each with a parental strand, and a new daughter strand.
Thus far no sugar-phosphate bonds have been cleaved. When replication is done each daughter ds DNA helix will have a single-stranded nick at the origin, and another one at the site where replication terminated, but those will be quickly repaired by DNA polymerase I and DNA ligase. Even if those two repair enzymes were inhibited, and you could somehow melt the DNA strands of both the daughter chromosomes, you would only be able to pull out the new DNA molecules (with the nicks), the two original parental strands are still topologically linked.
To resolve two concatenated DNA circles you need to make a double-stranded break in at least one of the ds circles. This enzymatic activity is provided by a class of enzymes named Type II DNA topoisomerases.