, makuri, leads to a two stage ko. If White cannot win this ko, then he can back out with a subsequently.
A connection at is highly undesirable for White. After
, the ko becomes a much larger direct ko.
The descent at is not good too, as it leads to a direct ko too. Or Black may play
at a, capturing two stones in sente.
The atari at is a failure. Instead of capturing at a, White can descend at
and there is nothing else Black can do.
The atari at here does not work either. The key move for White is
, which prevents the ko.
Without the black stone at a, is a mistake -
can extend and say arigato to Black for the free gift. Obviously White should not play
at b - Black will play
at
and thank White instead.
So without Black a, Black should atari at . The correct response is
, regardless of whether there is a Black stone at a.