The mathematical study of knots. Knot theory considers questions such as the following:
1. Given a tangled loop of string, is it really knotted or can it, with enough ingenuity and/or luck, be untangled without having to cut it?
2. More generally, given two tangled loops of string, when are they deformable into each other?
3. Is there an effective algorithm (or any algorithm to speak of) to make these determinations?
Although there has been almost explosive growth in the number of important results proved since the discovery of the Jones polynomial, there are still many "knotty" problems and conjectures whose answers remain unknown.
