This should be obvious, I think: The Second Amendment protects “arms,” and the D.C. v. Heller opinion discusses bows and knives as examples of such arms; opinions in the 1800s and 1900s dealing with state constitutional rights to bear arms also mention bladed weapons; and post-Heller opinions, such as from courts in Connecticut, Michigan, and Wisconsin agree. But some have disagreed — the Massachusetts government in the Caetano stun gun case before the Massachusetts high court, for instance, argued that Heller was limited to firearms. The New Jersey decision should be a helpful precedent, then, for other non-gun cases (though of course it doesn’t dispose of the question of exactly what weapons are protected, and where they can be possessed).
The Constitution Society has a handy document you might want to bookmark which covers this, as well as many other questions on related topics. In it, they go into a bit more detail about precisely what the Founders intended and what classes of weapons should be covered. (Emphasis added)
The U.S. Constitution does not adequately define “arms”. When it was adopted, “arms” included muzzle-loaded muskets and pistols, swords, knives, bows with arrows, and spears. However, a common- law definition would be “light infantry weapons which can be carried and used, together with ammunition, by a single militiaman, functionally equivalent to those commonly used by infantrymen in land warfare.” That certainly includes modern rifles and handguns, full-auto machine guns and shotguns, grenade and grenade launchers, flares, smoke, tear gas, incendiary rounds, and anti-tank weapons, but not heavy artillery, rockets, or bombs, or lethal chemical, biological or nuclear weapons. Somewhere in between we need to draw the line.