Can a wife return after her divorce of three days without a contract or mahr, especially if she has waived her rights before the return?

The divorce of the questioner's wife, at her request and with an agreement to waive her rights, could involve two scenarios:

1. The divorce is a khulʿ (dissolution of marriage initiated by the wife) in exchange for her waiver of rights. In this case, the wife cannot be taken back except with a new marriage contract. 2. The divorce is a revocable ṭalāq (repudiation) not in exchange for the waiver, and then the wife waived her rights after the divorce. In this case, if it was the first or second divorce, the husband has the right to take her back as long as she is within her waiting period (ʿidda).