Determining the sharpest sword in the world is challenging, as sharpness can depend on various factors, including the quality of the blade, the skill of the swordsmith, and the sharpness achieved through sharpening techniques. However, there are several swords known for their exceptional sharpness and cutting ability. Here are a few notable examples:

1. Japanese Katana: Traditional Japanese katana swords, known for their exceptional craftsmanship, are renowned for their sharpness. They are characterized by a single-edged curved blade and a sharp cutting edge, making them highly effective in cutting techniques.
2. Gurkha Kukri: The Gurkha kukri is a traditional Nepalese knife with a curved blade. It is renowned for its sharpness and versatility. The inwardly curved blade shape allows for powerful chopping and slashing strikes.
3. Damascus Steel Swords: Swords made using Damascus steel, a historical type of steel known for its strength and sharpness, are often highly regarded. Damascus steel swords can possess exceptional cutting ability due to their intricate pattern-welded construction.
4. Viking Ulfberht Swords: Ulfberht swords were high-quality Viking swords produced during the Viking Age. These swords were known for their exceptional sharpness and superior craftsmanship compared to other swords of the time.

It's important to note that the sharpness of a sword also depends on proper maintenance, regular sharpening, and the skill of the user in maintaining a keen edge. The sharpness achieved through sharpening techniques and regular upkeep is crucial in determining the cutting ability of a sword.