This ( and ) is not necessarily the continuation, but by playing the hane away from the position, White limits the loss to .
White's natural moves are good moves: blocking at , and either sacrificing in order to make shape as in this diagram, or extending as in the next diagram:
If the surrounding position is strong enough, White can extend (stretch).
This is unlikely to occur, however, for if White were strong around, Black should not want to cut through the keima anyway.
togo: Connection in sente I would say. The shape does not look very spectacular to me.
is a sacrifice for shape, sente, and aji.
togo: Black gets a second eye in sente, if I see correctly.
is a sacrifice, mainly for territory.
togo: Connects in sente, I would say.
The Japanese term tsuke-koshi is literally attach across. For some time we have used waist cut but this is not necessarily standard terminology.
In Chinese, alone is called 跨 (kuā), the -- sequence is called 跨断 (kuā duān). The character 跨 (kuā) means "straddle", and the character 断 (duān) means "cut".