Well, it is so surprising as being an HR how someone can be so irresponsible for preparing an appointment letter. Anyways, as notice period is not mentioned in the appointment letter you are not suppose to serve the notice period but if the HR is not responsible person and have done the blunder then it does not mean you do the same way. So, being a responsible employee better if you properly hand over the task or complete your tasks at the level so that it can be hand over to the other person, complete the formalities in a positive manner before leaving the company.