?What are Sangathis?

Definition of Sangati :

A particular variation of a phrase in a kriti or song with slight or large changes in tune and gamakas. Each phrase maybe repeated with different variations or sangatis.

Sometimes, in a kriti, the first lines will be varied systematically as the singer repeats it over and over. (This happens even in some movie songs). That is, the Pallavi will be sung intentionally in different melodies as it gets repeated. The text of the line will be the same, but the music will be different. Such variations are called 'Sangatis'.
If you listen to a Kriti like 'Vatapi Ganapatim' in Ragam Hamsadhwani, you will notice that the singer spends a long time mulling over the first line itself. But if you paid close attention to details, the singer is in fact varying the melody of that line over and over - in other words, the singer is producing the various 'Sangati's. Sometimes, after all the variations are done, the final version of the Pallavi could sound totally different from what one started out with. In fact, usually, the final version of the Pallavi is what is the official version of the Pallavi from that point on - and this is the version that is sung after each stanza (charanam) subsequently. Even Anu pallavi can have associated Sangatis.

It is said Saint Thyagaraja is credited with introducing sangati-s into the kriti form. If you look at the text of Muthuswamy Dikshitar krithis in the Sangita Sampradaya Pradarsini, very few sangatis are given for most of the compositions. However, when we listen to some of them sung these days, we hear lots of sangatis. It suggests that sangatis have been added by musicians at a later stage, but they have been careful about the lyrics and meanings.