1.2 Dense, Straight, Superstraight, and Simple Lacings 5

across during one of these journeys in the order that we visit them. We call an n-

lacing simple if the sequences of numbers corresponding to the associated journeys

are both nondecreasing. All the n-lacings we have discussed so far are simple n-

lacings. The lacing in Figure 1.7 is an example of a 3-lacing that is not simple.

Figure 1.7. A 3-lacing that is not simple and the two ways to travel from A1 to A3.

We end this chapter with three lemmas that summarize some basic properties

of n-lacings.

Lemma 1.1 (Verticals). Let l be an n-lacing. Then the following hold:

1. The number of verticals of l in column A equals the number of verticals in

column B.

2. The maximal possible number of verticals in l is n for n even, and n - 1 for n

odd.

Proof. Let a, b, and d denote the number of verticals in column A, the number of

verticals in column B, and the number of diagonals, respectively. Remember that

any of the eyelets can be endpoint of at most one vertical and that any of the

eyelets is endpoint of at least one diagonal. Therefore, there are exactly 2a eyelets

in column A in which exactly one diagonal ends and n-2a eyelets in which exactly

two diagonals end. Consequently, we can count the number of diagonals as follows:

2a + 2(n - 2a) = d = 2b + 2(n - 2b).

Hence, a = b. This proves the first part of this lemma. To prove the rest is a

straightforward exercise. ✷

Lemma 1.2 (Existence of Superstraight Lacings). Superstraight n-lacings ex-

ist only for even n.

Proof. The serpent n-lacings can be realized for any even n, which means that super-

straight n-lacings exist for even n. A superstraight n-lacing contains 2n segments,

n of which are horizontals and the remaining n are verticals. As a consequence of

the previous lemma, this number of verticals is even. ✷

Lemma 1.3 (Horizontals in Simple Lacings). Every simple n-lacing contains

the top horizontal and the bottom horizontal.

This lemma is an immediate consequence of the definition of simple n-lacings.