Basically they work like a sword. You, simply put, clobber people with it. Or you skewer them with it. You do not saw them with it. You need teeth for proper sawing.
And I don't think that, even with adamantine, you could cut people in twain. It's amusing, yeah, but not very viable. There's more to it than just the sharpness/wobbliness of the blade. Maybe you could cut someone in half if you held them taut on something like a vertical torture rack.
Also, having a "razor" edge on a sword is a waste and useless.
Alsoalso I doubt it is possible to have anything "a" molecule thick. But then I'm not a scientastic person. How thick is a molecule?
A katana is a sawing weapon, Really. It has microscopic serrations. Unlike a claymore, longsword, etc, a katana is "Drawn" over the object to be sliced, much like a good kitchen knife. (Western swords "Cleave", like a meat cleaver.) Katana are not used to strike one another in the fashion that a western fencing sword is. Katana have 2 distinct types of metal used in construction, and the cutting edge is VERY VERY FRAGILE. (A curse from feudal japan is "May your sword shatter and break") Katana have only 1 cutting edge. The other is blunt, and made of mild, low carbon steel. When using the katana to deflect a blow, this blunt side is offered to the incoming blade, in the attempt to shatter the enemy blade. This 2 metal composition is what causes the sword's unique bowed shape. (The two metals have different modulus of elasticity, and different rates of thermal contraction.)
Due to the fragility of high carbon steels, a katana needs to be "Folded" and "Laminated". Basically, this means that part of the structural integrity of the blade is maintaned by the softer, mild steel in the back part of the blade, because it is folded in with the high carbon, fragile steel in layers. The mild steel bulks up the blade, but makes it so that it wont shatter if it strikes poorly, and gives it a little flexibility it otherwise wouldn't have.
Imaginary Adamantine does not suffer a high fragility, like high carbon steel. As such, it wouldnt need lamination, could be made absurdly thin, and could be used absurdly fast. It would be like giving "Super paper cuts" rather than "Chopping".
Basically, a super paper-thin nano-serrated razorblade that never goes dull, and is 5ft long. Due to not needing to be laminated, and not needing a deflecting edge because the cutting edge cant shatter, it would\could be a 2-edged straight sword. (Though curved would permit easier use.)