Explanation:

- The Chalukya Dynasty, known for its significant contributions to Indian art and architecture, predominantly ruled the region of Vatapi.

- Vatapi is known today as Badami, which is located in the Indian state of Karnataka.

- Option 1: Kerala - While an influential region, the Chalukyas are not historically linked to Kerala.

- Option 2: Gujarat - This region was prominent in other historical contexts but not under the Chalukya reign of Vatapi.

- Option 3: Karnataka - This is the modern-day state where Vatapi (Badami) is situated.

- Option 4: Tamil Nadu - Although neighboring Karnataka, it was not where the Chalukyas established their rule.