The rarest katana in the world is difficult to determine definitively, as it depends on various factors such as historical significance, craftsmanship, provenance, and unique features. However, there are a few notable katanas that are considered exceptionally rare and highly prized:

1. Honjo Masamune: The Honjo Masamune is one of the most famous and revered swords in Japanese history. Believed to have been crafted by the legendary swordsmith Masamune, this katana is considered a national treasure of Japan. It is known for its exceptional craftsmanship, beauty, and historical significance. However, the Honjo Masamune is currently missing and its whereabouts are unknown, adding to its mystique and rarity.
2. Kogarasu Maru: The Kogarasu Maru is an ancient tachi (an older form of the katana) that is considered an Important Cultural Property of Japan. It is known for its unique crescent-shaped tip and is attributed to the famous swordsmith Amakuni, who is believed to have lived in the late 7th century.
3. Yasutsuna Ichimonji: The Yasutsuna Ichimonji is a renowned katana crafted by the swordsmith Ichimonji Yasutsuna, who lived during the late Heian period (11th century). Swords attributed to Yasutsuna are highly sought after for their exceptional craftsmanship and historical significance.
4. Fudo Masamune: The Fudo Masamune is another sword attributed to the legendary swordsmith Masamune. This sword is renowned for its quality and is considered one of Masamune's finest works. It is a national treasure of Japan and is currently held in the Tokyo National Museum.

These katanas represent only a few examples of exceptionally rare swords, and there are many other unique and historically significant katanas in existence. The rarity and value of a katana are influenced by various factors, including historical importance, craftsmanship, condition, and provenance. The scarcity of certain swords can also contribute to their rarity, as many ancient swords were lost or destroyed over time.