4 1 Setting the Stage
It is an easy exercise to check that there is exactly one bowtie n-lacing if n is even
and exactly (n+1)/2 different bowtie n-lacings in the case that n is odd. Figure 1.6
shows the different bowtie n-lacings for n = 2, 3, 4, 5, and 6.
Figure 1.6. The bowtie n-lacings for n = 2, 3, 4, 5, and 6.
Note that for fixed n all bowtie n-lacings have the same length. Furthermore,
by Lemma 1.1 below, a bowtie n-lacing contains the maximum number of verticals.
The serpent n-lacings clearly exist only for even n and the zigsag n-lacings exist
only for odd n. For n 2 both serpent and zigsag n-lacings come in pairs, one
member of the pair being the vertical mirror image of the second. For n = 2 the
bowtie and serpent n-lacings coincide. It should be clear from the diagram how to
construct the n-lacing representatives of the serpent and zigsag lacings.
1.2 Dense, Straight, Superstraight, and Simple Lacings
We call a lacing dense if it does not contain any verticals. This means that a dense
lacing zigzags back and forth between the two columns of eyelets. The crisscross,
zigzag, and star n-lacings are dense, whereas the bowtie, serpent, and zigsag n-
lacings are not.
We call an n-lacing straight if it contains all horizontals. Examples of straight
n-lacings are the zigzag, star, serpent, and zigsag lacings. We call an n-lacing su-
perstraight if it is straight and all nonhorizontal segments are verticals. The serpent
n-lacings are examples of superstraight n-lacings.
Clearly, we can travel from eyelet A1 to eyelet An along an n-lacing in exactly
two different ways; see Figure 1.7. Let’s note the indices of the eyelets we come