Squeeze tesuji in joseki

Does Black need to defend the cut at a?  

The question is whether Black needs to defend against the cut at a. Superficially this may look like a ladder question, but...

White cuts  

Black has a clever defense if White cuts. He starts with an atari at and then plays a net at . There is no escape at a or b.

White tries to break free  

White plays atari at , and again at . The plan is to break free at b if Black a.

Black squeezes  

However Black now plays a squeeze tesuji at .

Black squeezes  

After , it is clear that the white cut didn't work.

Reference position  

For reference, the starting position is the result of this rather common san-san invasion under the black hoshi stone.

Net  

Another reply to the white cut is the net of . This also captures , but the firmer capture through the squeeze tesuji is better. See Lessons in the fundamentals of Go.

Ladder  

The ladder is an even worse way to capture . Generally, capturing in a net (or another direct method such as the squeeze) is better than capturing in a ladder, because a ladder allows the opponent to play a ladder breaker, basically getting a free move elsewhere. See Net versus ladder.