Twisted four in the corner

One outside liberty  

If there are fewer than two outside liberties, White can turn this position into a ko (see below). prevents that and lives with full points. (Black playing at a also does that, but leaves a ko threat, whereas leaves none.)

White starting a ko  

If is tenuki, then at kills. After , White can finish the ko by filling at . If Black has no outside liberties, then would be better, since filling at would leave a ko threat. After Black retakes the ko with , Black can finish the ko with to live with full points. (The same applies if is tenuki.)

Note that after , would be either self atari or suicide. (The latter is if there are no outside liberties.)

Two outside liberties  

If there are at least two outside liberties, Black is alive with three points. (Three points instead of four because Black eventually needs a reinforcing move inside when White fills up one of the outside liberties.)

White cannot make a ko  

Thanks to the two liberties, Black can play atari at . Without one of those liberties, he would be putting himself into atari.

at  

When Black has exactly two outside liberties, White can fill either of them in sente. The threat is at least a ko to kill Black's group.

However, the indicated lines might be better than filling immediately, since in both indicated lines, and and each threaten to kill outright, rather than just threatening a ko to kill.