Some questions to the history experts here:
What kind of two handed swords existed around 1220? The German greatsword, commonly called the Zweihander, didn't appear until a couple of hundred years later, if I understand correctly, and it was mainly used against heavy armour and pike formations.
And about Muslim cavalry, didn't they carry curved sabres?
Finally, Greek Fire. Did it exist in the middle ages? And how was it used?
Cheers,
Eirik
You have right. Greatswords and even longswords weren't invented yet:
en.wikipedia.org/wiki/Longsword#_ref-7
en.wikipedia.org/wiki/Zweih%C3%A4nder
(a typical mistake in many rpgs inclung ArM calling the one-handed sword to longsword)
I'm not sure about Muslim cavalry but if I remember well I heard they used swords and not sabres.
But Muslim Turks used sabres so it depends.
Greek fire existed. I think wiki is quite detailed:
en.wikipedia.org/wiki/Greek_fire
It seems the correct name for the "longswords" of this period is the "arming sword", which is basically an about 1 meter long, single handed sword. Longsword would appear from time to time, I guess, but they were not in common use. The wikipedia article says that the longsword's early use would be into the 13th century.
I guess even greatswords would appear from time to time in the 13th century, but with the lack of plate armor, there would be no reason to wield it. But I guess some warriors just need that big sword to swing around. 
E.
The Muslims (Turkish or otherwise, infantry or cavalry) are mostly using curved swords at this point, popularised after the Seljuk conquests of the 11th century. That said, there are still a reasonable number of straight or only very slightly curved swords around, so personal preference (of the soldier or the employer) still plays a significant role.
You would also tend to find broadswords (such as the traditional viking sword), falchions (glorified meat cleaver), and messers (basicly sword sized knifes).