The correct answer is option 1. Amarasimha was a famous lexicographer of ancient India. A lexicographer is someone who compiles dictionaries. Amarasimha is known for his work called "Amarakosha," which is a Sanskrit thesaurus. This work is highly regarded and widely studied in the field of Sanskrit literature.

While option 2 suggests that Amarasimha was a Rajput king of the Chalukya lineage, there is no historical evidence to support this claim. It is important to note that Amarasimha is primarily known for his work as a lexicographer and not as a king.

Option 3 mentions that Amarasimha is a poet mentioned by Kautilya in his Arthashastra. However, there isn`t any reference or mention of Amarasimha as a poet in the Arthashastra by Kautilya. This option is incorrect.

Option 4 suggests that Amarasimha is a dance form of Odisha. This is incorrect as well since there is no dance form named Amarasimha associated with Odisha.

In conclusion, option 1, which states that Amarasimha is a famous lexicographer of ancient India, is the