The question asks about the commonalities between two historical places, Ajanta and Mahabalipuram.

Option 1 states that both Ajanta and Mahabalipuram were built in the same period. But it`s incorrect because Ajanta Caves were established around the 2nd century BCE to about 480 CE, while the monuments at Mahabalipuram were built in the 7th and 8th century AD.

Option 2 claims that they belong to the same religious denomination. This is not true. Ajanta caves are mainly Buddhist, while the temples of Mahabalipuram are Hindu.

Option 3 suggests that both have rock-cut monuments. This statement is correct because both Ajanta and Mahabalipuram are renowned for their rock-cut architecture.

Option 4 claims that none of the statements given above are correct. Since option 3 is correct, this statement is incorrect.

Therefore, the correct answer is Option 2, indicating that the common facet between Ajanta and Mahabalipuram is their rock-cut monuments.