In Arrian's account of this incident, he uses the word 'xiphos' which just means 'sword', and seems to have been a straight bladed, double edged weapon.

There are a few different versions of this story, some of the ancient sources. Aristobulus believed that Alexander just pulled the pins out of the knot and this exposed the ends of the chord so that it could be unravelled easily (this version is also preserved in Arrian, Anabasis, 2.3). Hope this helps!