Your suggested English sentence, "I know a son of K", is equivalent in English to "I know one of the sons of K".
But that is not what the Czech sentence says. The Czech implies that K has one son. (Or am I mistaken?)
The correct English is therefore "K's son" = "the son of K'. English idiom requires the "the" if you use the latter form.
A funny notion :)
Seeing "Znám Kateřiny syna", I still immediately interpret it as "I know Kateřina's son" (or.. the son of Kateřina) -- just with an unusual word order, but not unheard of. The interpretation comes from the fact that it's a much more likely thing to say. It also sounds like a lazy way (Czechs do this sometimes) of saying "Znám Kateřinina syna", i.e. replacing the possessive adjective with a plain genitive without switching the word order.
If I really wanted to say "I know the son's Kateřinas" (let's say he's dating more than one girl named Kateřina), I would say "Znám synovy Kateřiny", using the possessive adjective, to remove the other meaning completely.