A tie knot is most easily untied by pulling the passive (thin) end out through the knot. It may be readily observed that the resulting conformation, when pulled from both ends, yields either the straightened tie or a tie with a subsequent smaller knot. More formally, when the passive end is removed and the two tie ends joined, the tie may either be knotted or unknotted. Any conformation that can be continuously deformed to a standard ring (the canonical unknot) is said to be unknotted.
We first note that a tie knotted up to but not including the terminal sequence corresponds, upon pulling out the passive end, simply to a string wound in a ball with the interior and exterior ends protruding. Since the ball can be undone simply by pulling the exterior end of the string, all such conformations are unknotted. The terminal sequence, in particular the move T (through), is responsible for any remaining knot. This can best be observed diagrammatically by projecting the knot on to the plane. In Fig. 4, the solid spheres represent the non-terminal sequences (which cannot give rise to a knot), with the terminal moves of the active end drawn explicitly. The dotted lines represent imaginary connections of the tie ends and, for the purposes of our argument at least, cannot be cut. The left diagram, with the terminal sequence , can be continuously deformed to a loop and is hence unknotted. No amount of deformation of the right diagram, with terminal sequence , will reduce the number of intersections below three. It is the simplest knotted diagram, a trefoil knot.
Figure 4: The left diagram, with terminal sequence , is unknotted, while the right, with terminal sequence , forms a trefoil knot.
We find, then, that a tie knot will straighten upon pulling out the passive end if the terminal sequence is and remain knotted otherwise. Note that the number of unknotted knots of length h corresponds exactly to walks in (7).