I don't have any games to show, because this was just a question that I thought about and couldn't get my head around.

It's very important to look at the SPECIFIC situation. It's very difficult to answer your question without a specific position.

Of course, Ed is right, that without specifics, it's hard to give an adequate answer. But I will toss out a few generalizations anyway.

You are right that both the one and two space jumps can be cut. That doesn't necessarily mean that they will be cut.

First of all, it depends a great deal on where the surrounding stones are. If the jump is in the middle of nowhere, trying to cut without support may very well leave the cutter at a disadvantage.

If you are thinking about jumping in order to get out into the center, then you might also look at the alternatives. Both the iron pillar and the diagonal move can not be cut, but they both move slower towards the center. If you end up getting surrounded, then it doesn't help you much that you can't be cut. Ideally, when you jump out, you are not just running, but also threatening something, so that when your opponent defends, you will have time to secure your stones. Also, if your jump does get cut, it will be in the process of a fight, and not the end of the story. Presumably you will get something in return, and if you don't, then you probably shouldn't have been in that situation in the first place.

If you are thinking about jumping to attack, for example in order to surround your opponent, then being cut is a two way street. Your opponent doesn't only need to cut you - he also needs either to live, or to capture some of your cutting stones.

In any case, try posting an example, and you'll see why there's usually no clear cut answer.

I will try to find a game with a one or two space jump.

The following are all 2-space jumps

It'd be quite interesting to find a game without any.

That's a good idea, and since all games have jumps, as EdLee pointed out, I'll post one of my next games here then.

Since both jumps can be cut, especially the two space jump, why would it be a good way of fleeing or attacking? In what sort of a position would these jumps be safe from cutting?

I have been reading Opening Theory Made Easy for a few days, and it uses one space and two space jumps to run away/attack a lot. Since both jumps can be cut, especially the two space jump, why would it be a good way of fleeing or attacking? In what sort of a position would these jumps be safe from cutting?

I don't have any games to show, because this was just a question that I thought about and couldn't get my head around.