Assuming a "standard" LP/HP jet engine (say that of a B737 NG), what is the relationship between N1 and N2? As there is no direct linkage between the two shafts the ratio between N1/N2 won't be constant.
You answered your own question. There is indeed no fixed relationship.
Is there a way to calculate N1 from N2 (if the parameters such as air density, temperature etc... are known)?
Yes, if you have access to either the engine model (or "deck" as its often called, after the "decks" of cards used to calculate it in days of yore) or engine "running line" data (basically, the steady state outputs from the deck). That would be for the specific engine type/variant (and, to be very precise, data for the actual engine in question)
That would be valid for a stable engine condition. Under acceleration the relationship would be subtly different, but could be modelled if one knew the time history of the other parameters, rather than a single set of conditions.