Knot a handkerchief
If I give you a handkerchief or scarf, and ask you to hold one end in each hand, and tie a knot in the handkerchief without letting go of the ends, could you do it? In fact, you can - but it requires a little bit of a cheat!
The shape your arms and the handkerchief make is a circle, which can never become a circle with a knot in it, unless you break the circle at some point.
The key to the trick is this: you must first pass the handkerchief to someone else, so that they are holding both ends.
Then, fold your arms. You must make sure that your arms are folded in such a way that one hand sits on top of the arm it rests on, and the other arm is underneath.
Now, grab the two ends of the scarf back from your friend, whilst keeping your arms folded. You will find that you can then easily tie a knot without letting go, by simply uncrossing your arms and pulling the ends apart!
If you like this trick you may enjoy the book: Martin Gardner, Mathematical magic show, Penguin Books, 1985