I find many people from Karnataka and TN arguing as to where exactly is HK Falls. Being a Kannadiga I am proud to say that the name of HK Falls is derived from "Hoge" - smoke, and "Kallu" - stones, i.e., smokes emanating from stones. But, let me quote the bare facts now:
Cauvery originates in Karnataka at Talakaveri and exits the state near Muthathi in Chamarajanagar district at the Latitude/Longitudes: 12N12, 77E34. It stays in TN for a very short route from this point (which can be accessed from Pennaguram in Dharmapuri district in TN) upto 12N6'40" 77E46'30". It then re-enters karnataka for some short time. It is at this boundary between Karnataka and TN, that Hogenakkal Falls exits. Hence HK Falls belongs equally to both states (it can be approached from both states). The north part of the Falls is in TN, while the southern and majority part of the HK Falls is in Karnataka. Cauvery again re-enters TN at 11N54'37" 77E45'20", which is where stanley reservoir is built.
Hence, the bottom line is that HK Falls EQUALLY belongs to Karnataka and TN, or rather it is a National Property. Hence, TN doesnt have any exclusive right to build dams at this point, since water which flows later from this point re-enters Karnataka and affects farmers in this region of Karnataka.