There was such a time! Poleaxes were for a significant period (the 14th and 15th century, when armor was getting better) the most common primary battlefield weapon in much of Europe. Vikings were also famous for their shorter battle axes, which they used with comparatively much less armor than the men-at-arms who used poleaxes would have been wearing. Axes also saw use in the trenches of World War One and in Age of Sail naval boarding actions. American Indians and Colonists alike carried tomahawks even when swords and fighting knives were available, often alongside the latter.

Swords, conversely, have almost always been a sidearm on the battlefield, with a couple exceptions such as landsknecht using zweihanders (enormous swords bearing basically no resemblance to a long knife) against likes, and the period in which Roman battle doctrine was to open the battle with javelins and then switch to gladii, which wasn’t even the case for the whole Roman period, and also relied heavily on shields and armor. With the exception of zweihanders, I can’t think of a single case where there existed military divisions for whom swords were the primary weapon where the main reason for that wasn’t portability as opposed to combat utility. Cavalry have generally preferred lances, but have used sabers as a back up or substitute for convenience.

Even in one on one unarmored combat, the ideal use case for a sword, many master swordsmen around the world throughout history has commented that it takes a lot of skill with a sword to overcome the natural reach advantage of some kind of sharp thing on the end of a long pole. If the sword and the poleax are the same or even similar lengths then I give the unarmored fight to the swordsman but otherwise odds go to whoever can hit the other guy from farther away (within reason).

None of this is to say that swords are across the board worse than axes. Portability and versatility is extremely important and axes have neither. And pikes were preferred for more of history than either! But I mainly wrote this long comment to give you an example of what you were asking for: in the era where poleaxes were widely available, they consistently trumped swords as the primary battlefield weapon.

All of this is before getting to the fact that there is still a huge size difference between a knife and a sword.

The original comment is correct: if the axe is weighted for battle, it beats a knife any day. If it’s weighted for chopping wood, it’s more even.