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Size of Knots

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 tex2html_wrap_inline948 , our initial constraint. Let tex2html_wrap_inline984 be the number of walks beginning with tex2html_wrap_inline948 and ending with tex2html_wrap_inline988 , tex2html_wrap_inline990 the number of walks beginning with tex2html_wrap_inline948 and ending with tex2html_wrap_inline994 , etc. Accordingly, since at any given site the walker chooses between two allowed steps,

  equation96

Since the only permitted terminal step sequences are tex2html_wrap_inline996 and tex2html_wrap_inline998 , we are interested in the number of walks of length n = h-2 ending with tex2html_wrap_inline988 or tex2html_wrap_inline948 , after which the respective remaining two terminal steps may be concatenated.

We begin with the derivation of tex2html_wrap_inline1006 . Now tex2html_wrap_inline948 can only follow from tex2html_wrap_inline988 and tex2html_wrap_inline994 upon each additional step, that is,

  equation115

from which it follows that

  equation121

Combining (2) and (4) gives rise to the recursion relation

  equation130

With initial conditions tex2html_wrap_inline1014 and tex2html_wrap_inline1016 (since no consecutive steps can be in the same direction), (5) is satisfied by

  equation139

The recursion relation for tex2html_wrap_inline984 is identical to (5), but with initial conditions tex2html_wrap_inline1020 and tex2html_wrap_inline1022 . Accordingly,

  equation150

The number of possible tie knots of half-winding number h is equal to the number of walks of length h-2 ending in tex2html_wrap_inline988 or tex2html_wrap_inline948 , that is,

  equation157

where K(1) = K(2) = 0. For tex2html_wrap_inline1034 , 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 tex2html_wrap_inline772 ) 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).


next up previous
Next: Shape of Knots Up: Tie Knots and Random Previous: Tie Knots as Random

Yong Mao
Sat Nov 7 16:03:57 GMT 1998