Knots

A knot is a closed curve of finite length in ℝ3 that does not intersect itself. Two knots are equivalent if one can be deformed into the other by moving it around without passing one strand through another. (We will define equivalence more formally in the next section). One of the fundamental problems in knot theory is to determine whether two knots are equivalent. If two knots are equivalent, then this can be demonstrated: For example, we could produce a video showing one knot being continuously deformed into the other. Unfortunately, if two knots are not equivalent, then it is not at all clear how to prove this. The main approach to this problem has been the development of knot invariants: values associated with knots such that equivalent knots have the same value. If such an invariant takes different values on two knots, then the two knots are definitely not equivalent.