This was the title of a great lecture held by Gareth Williams two days ago. The lecture was held using Elluminate and about 60 students attended. In my opinion this is a large number of students, since this lecture was not part of the course or so ... it was held for those who were interested in why we need topology. ;)

But is the question "Can one knot untie the other?" important at all? Is it only theoretical or even practical?

Assume we are given a piece of string with a knot tied in it. Can we then tie another knot which will combine with the first knot to produce an unknotted piece of string? Answering this question is not only interesting it is of practical interest for sailors, biologists (DNA), bungee-jumpers etc. as well.

After considering Seifert's surfaces, adding some knots we then came to the conclusion that one knot cannot be tied after another in such a way as to make the result equivalent to the trivial knot, and so, one knot does not untie another. :)