The sharpest sword in the world would likely be one that has been expertly crafted with high-quality materials and honed to an exceptionally fine edge. Achieving an extremely sharp edge depends on several factors, including the quality of the steel, the blade geometry, and the skill of the swordsmith.

While it is challenging to identify a specific sword as the "sharpest" due to the subjective nature of sharpness and the absence of universally accepted measurements, some types of swords and blade styles are known for their exceptional sharpness. Here are a few examples:

1. Japanese Katana: The traditional Japanese katana is renowned for its sharpness. Through the meticulous process of folding and differential hardening, the katana's cutting edge can be honed to an incredibly sharp level, allowing it to achieve precise and effective cuts.
2. Damascus Steel Swords: Swords made from Damascus steel, whether traditional or modern, are often associated with remarkable sharpness. The layering and folding techniques used in forging Damascus steel can result in blades with an exceptionally keen cutting edge.
3. Straight Razors: While not swords in the traditional sense, straight razors are designed specifically for shaving and are known for their extreme sharpness. Crafted from high-quality steel and honed to a fine edge, straight razors can achieve an exceptional level of sharpness.

It's important to note that achieving and maintaining an incredibly sharp edge requires regular maintenance, proper sharpening techniques, and care in handling. The sharpness of a sword can diminish over time with use, and regular maintenance and sharpening are necessary to keep the blade at its sharpest.

Ultimately, the sharpness of a sword is influenced by various factors, including the materials, craftsmanship, and maintenance. Additionally, personal preferences and intended use can influence one's perception of sharpness.