Here is a thread on a high quality Mongol saber (
http://forums.swordforum.com/showthr...ghlight=mongol).
In general Mongol swords resembled those of the people they conquered. So the eastern empire had swords with Chinese and Tibetan influences such as the one above, and the Ilkhnids had swords resembling the shamshir.
The only 13th c. blade I have seen was from Persia and looked like the one in the link provided, so that may be the older design, with the raised yelman.
Often it is stated that the straight Chinese sabers gave way to curved ones with the influence of "step" peoples that would include the Mongols.
So it seems that curved sabers may have spread from China to Poland with the spread of the Mongols, but with local designs influencing what was actually used by the Mongols in a particular region.
Josh