E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
A great difficulty. Gordius, a peasant, being chosen king of Phrygia, dedicated his waggon to Jupiter, and fastened the yoke to a beam with a rope of bark so ingeniously that no one could untie it. Alexander was told that whoever undid the knot would reign over the whole East. Well then, said the conqueror, it is thus I perform the task, and, so saying, he cut the knot in twain with his sword.
To cut the knot is to evade a difficulty, or get out of it in a summary manner.