If you look at historical records like the Bayeux Tapestry you will see that the mounted troops used spears (lances came later) and did not carry swords. If you click on the link I provided there are a couple of videos that go into this. Swords have the glory but it’s been the spear that’s been the premier hand weapon on the battlefield since time immemorial. Well into the age of gunpowder, cannon and arquebus were still protected by pikes. Spears are still in use to this day in the form of bayonets.
Now, if you’re talking single-combat then that’s a different matter.