The number of possible knots as a function of half-winding number, K(h), corresponds to the number of possible walks of length h subject to the initial and terminal conditions.
We begin by considering all walks of length n beginning with , our initial constraint. Let be the number of walks beginning with and ending with , the number of walks beginning with and ending with , etc. Accordingly, since at any given site the walker chooses between two allowed steps,
Since the only permitted terminal step sequences are and , we are interested in the number of walks of length n = h-2 ending with or , after which the respective remaining two terminal steps may be concatenated.
We begin with the derivation of . Now can only follow from and upon each additional step, that is,
from which it follows that
Combining (2) and (4) gives rise to the recursion relation
With initial conditions and (since no consecutive steps can be in the same direction), (5) is satisfied by
The recursion relation for is identical to (5), but with initial conditions and . Accordingly,
The number of possible tie knots of half-winding number h is equal to the number of walks of length h-2 ending in or , that is,
where K(1) = K(2) = 0. For , the sequence appears as K(h) = 0, 0, 1, 1, 3, 5, 11, 21, 43, yielding 85 knots in total. Note that for knots of odd h (those initiated by ) to end with the active end facing inside in, the knot must begin with the tie inside out around the neck, as an odd number of half-turns would suggest (Fig. 1).