On the structure of the five-point condition

The five-point condition has an interesting structure that is worth a closer examination. We have already seen some of its properties in the Section 4.1 where we highlighted the possibility of illustrating the structure of the five-point condition with the help of Hamilton cycles. In the following, we will see that there are redundancies in the five-point condition and how the five-point condition in rank 2 can be written with the help of determinants. We have already learned that zeros in the image of the phirotope cause the five-point condition to automatically evaluate to zero. Therefore, we will examine uniform phirotopes in this section.

Consider a uniform rank-2 phirotope ϕ on six elements E = {1, . . . ,6}. According to the Lemma 4.3, it is realisable, if and only if for each five-element subset the five-point condition holds true. There are six of such five-element subsets and each of them contains all but one indices. In the following, we denote by Υ_k the five-point condition that does not contain the index k. With Υ₆ = 0 we can realise ϕ|_[5]. For this we choose the pointsP₁,P₂, and P₃ that realise the indices 1, 2, and 3 in general position. The position of the points P4 and P5 that realise 4 and 5 are then determined. In the same way, Υ₅ yields the position of (again)P₄ and P₆, if we choose the sameP₁,P₂, andP₃ to realise 1, 2, and 3. Now the only thing that might still forbid the realisability of the whole phirotope is the value of ϕ(5,6). The corresponding points are already determined and for the phirotope to be realisable, ϕ(5,6) =ω(det(P₅, P₆)) has to hold true. This is for example ensured by Υ₄ = 0.

Note that the realisability ofϕwas decided using only the five-point conditions Υ₄, Υ₅, and Υ₆. Thus, Υ₁= 0, Υ₂ = 0, and Υ₃ = 0 have to follow from Υ₄= 0, Υ₅ = 0, and Υ₆= 0. This can be proven as follows.

Consider Υ₄, Υ₅, and Υ₆. The termJ4,5Kis only contained in formula Υ₆,J4,6Konly in Υ₅, and J5,6Konly in Υ₄. We write

Υ₆ =A·J4,5K−B, Υ₅ =C·J4,6K−D, Υ₄ =E·J5,6K−F,

whereA, . . . , F stand for polynomials of squared phirotope values that complete the five-point conditions and do not contain J4,5K,J4,6K, orJ5,6K. The terms can also be depicted as (parts of) Hamilton paths as introduced in the Chapter 4.1. Recall that an edge between the vertices labelled witha andbcorresponds toJa, bK. This means that A andB are graphs on the vertices in {1,2,3,4,5}. B corresponds to all Hamilton cycles not containing the edge (4,5), while A contains all (undirected) Hamilton paths from 5 to 4 or of length four, cf. Figures 4.2 and 4.3.

The five-point condition Υ₃ contains all three termsJ4,5K,J4,6K, and J5,6K. Each summand

4.3. On the structure of the five-point condition

Figure 4.2.: The Hamilton paths that correspond to the term A.

1 2

Figure 4.3.: The Hamilton cycles that correspond to the termB.

of Υ₃ contains either one or two of the terms J4,5K, J4,6K, andJ5,6Kand thus, we can write it as

One possible solution is given by

α=ACW +BCY +ADZ, β=AEV +BEX+AF Z, γ =CEU +DEX+F CY,

δ=AZ, =CY, ζ =EX, η=−ACE.

With this, all terms containing J4,5K,J4,6K, and J5,6Kcancel:

α·Υ₄+β·Υ₅+γ·Υ₆

+δ·Υ₄·Υ₅+·Υ₄·Υ₆+ζ·Υ₅·Υ₆ +η·Υ₃

=−ACF W −ADEV −ADF Z−BDEX−BCEU −BCF Y

= 0. (4.12)

The last line can be checked with the help of a computer algebra system, cf. Appendix B.

Another way of showing the dependence of the five-point conditions is transposing Υ₄ = 0, Υ₅= 0, and Υ₆ = 0 as follows

J4,5K= B

A, J4,6K= D

C, J5,6K= F

E, (4.13)

which can be done asϕ was assumed to be uniform. Inserting these into Υ₁, Υ₂, and Υ₃ will each yield zero (cf. Appendix C).

All in all, this finding fits the theoretical contemplation at the beginning of this section. If three five-point conditions evaluate to zero, this is indeed sufficient for the realisability of a uniform, non-chirotopal rank-2 phirotope on five indices.

The result can be generalised to phirotopes with larger index sets. Consider a uniform, non-chirotopal rank-2 phirotope on [n]. Here, we have ⁿ₅ five-point conditions, so the number of five-point conditions is of quintic order in n. For three elements a, b, c∈ [n], we choose their affine representatives to be the projective basis that determines the position of all other points.

Then, according to the considerations above, each pair of indicesx, y∈[n]r{a, b, c}has to be part of at least one five-point condition. Thus, we only need ⁿ⁻³₂ five-point conditions, namely those on {a, b, c, x, y}. This number is of quadratic order inn.

4.3. On the structure of the five-point condition

Note that the argument we have used can only be applied to rank 2. In rank 2 the five-point condition is a formula on five elements and thus has two elements more than a projective basis, which consists of three elements. So in rank 2, the phirotope value of every pair of indices appears in a five-point condition together with the indices of the chosen projective basis. In arbitrary rankd, the five-point conditions are formulae on d+ 2 elements. This number differs from the number of elements in a projective basis, which containsd+ 1 points, by only one element.

Another interesting structural property of the five-point condition is that it can be written in terms of determinants. To see this, consider a five-point condition in which again one bracket is factored out. For example, consider again Υ₆ =A·J4,5K−B. It holds true that

We want to identify in which cases the value of the term ϕ(4,5) is irrelevant for the realisability of the phirotope. This can only be the case, if the phirotope contains a zero in its image or is chirotopal. Otherwise, the value of the term ϕ(4,5) cannot be irrelevant. Changing it will also change the value of the five-point condition. We want to assume that the phirotope ϕ|_[5] is uniform because the Theorem 4.23 guarantees that the five-point condition Υ₆ holds true, if the phirotope contains a zero in its image. Thus, we deal with the chirotopal cases:

• ϕ is chirotopal.

We will show that in this case the last column of the determinant of the termAis a multiple of the second one. By assumption, none of the phirotope values equals zero, thus

cr_ϕ(1,2|4,5) =±1 ⇒ J1,4K·J2,5K=J1,5K·J2,4K,

cr_ϕ(1,3|4,5) =±1 ⇒ J1,4K·J3,5K=J1,5K·J3,4K, cr_ϕ(2,3|4,5) =±1 ⇒ J2,4K·J3,5K=J2,5K·J3,4K.

By applying Laplace expansion toA we obtainA= 0.

Note that, as we are dealing with rank-2 phirotopes, chirotopality implies that the phirotope is realisable. And indeed, if all cross ratios are real, the last two columns of the matrix in B will also be equal. Thus,B = 0 and the five-point condition is satisfied.

• ϕ|_{1,2,3,4} is chirotopal.

For symmetry reasons, this case is analogous to the case in whichϕ|_{1,2,3,5} is chirotopal.

If the minor ϕ|_{1,2,3,4} is chirotopal, then the cross ratio phase cr_ϕ(1,2|3,4) and the cross ratio phases of all permutations of the four indices are real. As none of the involved terms equals zero by assumption, this yields:

J1,3KJ2,4K=J2,3KJ1,4K J1,2KJ3,4K=J2,3KJ1,4K J1,2KJ3,4K=J1,3KJ2,4K

By multiplying the first equation with J1,2K, the second with J1,3Kand the third with J2,3K, we obtain that all 2-by-2-subdeterminants that consist of entries of the first two columns vanish:

Thus, A = 0. Note that the same calculations apply to B, meaning that if ϕ|_{1,2,3,4} is chirotopal, the subdeterminants of the first two columns ofB also vanish. This implies that if ϕ|_{1,2,3,4} is chirotopal, ϕ is always realisable, regardless of the valuesϕ(1,5), ϕ(2,5), ϕ(3,5), andϕ(4,5).

So, we have seen that chirotopality – even of only a minor – will result in the vanishing of the terms A andB.