I've been reading up on Knot Theory (which is not my area of expertise) and am stuck in the following bit of logic:
Statement 1: Every knot can be represented as a braid.
Statement 2: There's a straightforward way of doing this.
Statement 3: Dynnikov gives an algorithm for telling if two braid words are equivalent via computing the "Dynnikov Coordinates".
So it seems to me that in practice, if you have two knots and want to determine if they are equivalent, you turn them both into braids and compare Dynnikov Coordinates.
Is this right? If so, why isn't this considered the ultimate knot invariant? What obvious point am I missing?