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