It is general practice that one has to complete the probation period.
Generally every company incorporates the class in their agreement that a month or two prior notice has to be given by the aggrieved party or either one has to be compensated as per the agreement. ( two or three months salary)
As per your versions nothing such has been mentioned in the agreement.
Unless one go through your appointment letter it is hard to come to the conclusion.
Coming to the resignation point have you sent it via email (as you said having personal problem) or personally approached HR, if so have you obtained the acknowledgement for the same for acceptance / nonacceptance.
For instance, if your version is true i.e., no notice period is mentioned in the agreement and not accepting the Laptop/ Resignation, you should file a complaint at the local PS stating all the facts and give a complaint to the grievances officer. (Generally every company has a grievance cell) Don't forget to obtain the Acknowledgement
Even though you're not get justice you can approach the Labor court, industrial tribunal or state/central Administration tribunal under which your company has been established.
Good Luck / All the Best