All you need to know about Carpenter's Square - 13
Does the hane at first line help black?
is correct. Because of the hane at first line, black has the leisure to play at key point *b4*, resulting in a ko fight. may play first, getting exactly the same result.
can also create a ko fight. However black gets to take the ko first so this result is considered a failure for white.
in this diagram is at the wrong side. may also play directly. In any case, the missing hoshi stone is black's Achilles' heel.
How about *b2*? This is similar to Diagram 12.2 and does make any difference. So why is this diagram repeated here? At least one textbook I know mentions that external liberty is not a factor to consider in this shape. I would like to argue against that conclusion based on this diagram.
In Diagram 13.5 is key point making a T-four killing shape to prevent black from playing 'a'. If we modify the original problem a bit, giving black one liberty at each side, can play at the key point now and white cannot cut *w9*. Black would play later to form a comfortable seki.
So is correct. is to enlarge the eyespace. is important to reduce eyespace from the other side. has to make an eye to avoid Bulky Five so the result is a ko. If playing instead of *b4*, black would get the same result as in Diagram 13.2. However, one can see if black wins the ko, current diagram is slightly better off for black. That's what we take the extra miles to explain.
To summarise: there are multiple ways to form ko fights. Diagram 13.2 is the best for both sides. Diagram 13.7 may be better for black if there are additional liberties outside.