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Partitions of unity in \(\mathrm {SL}(2,\mathbb Z)\), negative continued fractions, and dissections of polygons

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Abstract

We characterize sequences of positive integers \((a_1,a_2,\ldots ,a_n)\) for which the \(2\times 2\) matrix \(\left( \begin{array}{ll} a_n&{} \quad -\,1\\ 1&{}\quad 0 \end{array} \right) \left( \begin{array}{ll} a_{n-1}&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \cdots \left( \begin{array}{ll} a_1&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \) is either the identity matrix \(\mathrm {Id}\), its negative \(-\,\mathrm {Id}\), or square root of \(-\,\mathrm {Id}\). This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.

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Notes

  1. This remark and examples were communicated to me by Alexey Klimenko.

  2. Conway and Coxeter worked with so-called frieze patterns (see Sect. 6), but the equivalence of their result to the classification of totally positive solutions of Problem II is a simple observation; see [4, 20].

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Appendix: Conway–Coxeter quiddities and Farey sequences

Appendix: Conway–Coxeter quiddities and Farey sequences

This section is an overview and does not contain new results. We describe the Conway–Coxeter theorem, formulated in terms of matrices \(M_n(a_1,\ldots ,a_n)\), and a similar result in the case of Problem III, obtained in [6]. We also briefly discuss the relation to Farey sequences.

In the seminal paper [7], Conway and Coxeter classified solutions of Problem IIFootnote 2 that satisfy a certain condition of total positivity. These are precisely the solutions obtained from the initial solution \((a_1,a_2,a_3)=(1,1,1)\) by a sequence of operations (1.2). Their classification of totally positive solutions beautifully relates Problem II to such classical subjects as triangulations of n-gons. Furthermore, the close relation of the topic to Farey sequences was already mentioned in [8]. It turns out that the Conway–Coxeter theorem implies some results of [14] about the index of a Farey sequence.

1.1 Total positivity

The class of totally positive solutions of Problem II can be defined in several equivalent ways. Coxeter [8] (and Conway and Coxeter [7]) assumed that all the entries of the corresponding frieze are positive.

Another simple definition is based on the properties of solutions of the Sturm–Liouville equation.

Definition 7.3

A solution \((a_1,\ldots ,a_n)\) of Problem II is called totally positive if there exists a solution \((V_i)_{i\in \mathbb {Z}}\) of the Eq. (1.6) that does not change its sign on the interval \([1,\ldots ,n]\), i.e., the sequence of n numbers \((V_1,V_2,\ldots ,V_{n})\) is either positive, or negative.

In the context of Sturm oscillation theory, this case is often called “non-oscillating,’ or “disconjugate.’ The index of the corresponding broken line is equal to \(\frac{1}{2}\), see Sect. 5.

Remark 7.4

Note that since \(M_n(a_1,\ldots ,a_n)=-\mathrm {Id}\), every solution is n-antiperiodic, so that it must change sign on the interval \([1,n+1]\).

Let us give an equivalent combinatorial definition. Consider the following tridiagonal \(i\times {}i\)-determinant

$$\begin{aligned} K_i(a_1,\ldots ,a_{i})= \left| \begin{array}{cccccc} a_1&{}\quad 1&{}\quad &{}\quad &{}\quad \\ 1&{}\quad a_{2}&{}\quad 1&{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad \ddots &{}\quad \!\!\ddots &{}\quad \\ &{}\quad &{}\quad 1&{}\quad a_{i-1}&{}\quad \!\!\!\!\!1\\ &{}\quad &{}\quad &{}\quad \!\!\!\!\!1&{}\quad \!\!\!\!a_{i} \end{array} \right| . \end{aligned}$$

This polynomial is nothing but the celebrated continuant, already known by Euler, and considered by many authors. It was proved by Coxeter [8] that the entries or a frieze pattern can be calculated as continuants of the entries of the second row.

It is also well known, see, e.g., [4] (and can be easily checked directly), that the entries of the \(2\times 2\) matrix (1.1) can be explicitly calculated in terms of these determinants as follows:

$$\begin{aligned} M_n(a_1,\ldots ,a_n)= \left( \begin{array}{cc} K_n(a_1,\ldots ,a_{n})&{}\quad -K_{n-1}(a_2,\ldots ,a_{n})\\ K_{n-1}(a_1,\ldots ,a_{n-1})&{}\quad -K_{n-2}(a_2,\ldots ,a_{n-1}) \end{array} \right) . \end{aligned}$$

The condition \(M_n(a_1,\ldots ,a_n)=-\mathrm {Id}\) implies that

$$\begin{aligned} \begin{array}{rcl} K_{n}(a_i,\ldots ,a_{i+n-1})&{}=&{}-1,\\ K_{n-1}(a_i,\ldots ,a_{i+n-2})&{}=&{}0,\\ K_{n-2}(a_i,\ldots ,a_{i+n-3})&{}=&{}1, \end{array} \end{aligned}$$

for all i.

The following definition is equivalent to Definition 7.3.

Definition 7.5

A solution \((a_1,\ldots ,a_n)\) of Problem II is totally positive if

$$\begin{aligned} K_{j+1}(a_i,\ldots ,a_{i+j})>0 \end{aligned}$$

for all \(j\le {}n-3\) and all i. Note that we use the cyclic ordering of the \(a_i\).

1.2 Triangulated n-gons

The Conway–Coxeter result states that totally positive solutions of Problem II are in one-to-one correspondence with triangulations of n-gons.

Given a triangulation of an n-gon, let \(a_i\) be the number of triangles adjacent to the \(i^th \) vertex. This yields an n-tuple of positive integers, \((a_1,\ldots ,a_n)\). Conway and Coxeter called an n-tuple obtained from such a triangulation a quiddity.

Theorem

(see [7]). Any quiddity of a triangulation is a totally positive solution of Problem II, and every totally positive solution of Problem II arises in this way.

A direct proof of the Conway–Coxeter theorem in terms of \(2\times 2\)-matrices is given in [4, 11]. For a simple direct proof, see also [16].

Example 7.6

For \(n=5\), the triangulation of the pentagon

generates a solution \((a_1, a_2,a_3, a_4,a_5)=(1, 3, 1, 2, 2)\) of Problem II. All other solutions for \(n=5\) are obtained by cyclic permutations of this one.

1.3 Gluing triangles

Obviously, every triangulation of an n-gon can be obtained from a triangle by adding new exterior triangles.

Example 7.7

Gluing a triangle to the above triangulated pentagon

one obtains the solution \((1, 3, 2,1, 3, 2)=(1, 3, 1+1,\,1,\, 2+1, 2)\) of Problem II, for \(n=6\).

An operation (1.2) applied to a quiddity consists in gluing a triangle to a triangulated n-gon, so that the Conway–Coxeter theorem implies the following statement (see also [11], Theorem 5.5).

Corollary 7.8

Every totally positive solution of Problem II can be obtained from the initial solution \((a_1,a_2,a_3)=(1,1,1)\) by a sequence of operations (1.2). Conversely, every sequence of operations (1.2) applied to this initial solution is a totally positive solution of Problem II.

For a clear and detailed discussion; see [4].

1.4 Indices of Farey sequences as Conway–Coxeter quiddities

Relation to Farey sequences and negative continued fractions was already mentioned by Coxeter [8] (see also [22]).

Rational numbers in [0, 1] whose denominator does not exceed N written in a form of irreducible fractions form the Farey sequence of order N. Elements of the Farey sequence, \(v_1=\frac{a_1}{b_1}\) and \(v_2=\frac{a_2}{b_2}\), are joined by an edge if and only if

$$\begin{aligned} |a_1b_2-a_2b_1|=1. \end{aligned}$$

This leads to the classical notion of Farey graph. The Farey graph is often embedded into the hyperbolic plane, the edges being realized as geodesics joining rational points on the ideal boundary.

Fig. 1
figure 1

The Farey sequence of order 5 and the triangulated hendecagon

The main properties of Farey sequences can be found in [15]. A simple but important property is that every Farey sequence forms a triangulated polygon in the Farey graph. A Conway–Coxeter quiddity is then precisely the index of a Farey sequence, defined in [14].

The Conway–Coxeter theorem implies the following.

Corollary 7.9

A solution \((a_1,\ldots ,a_n)\) of Problem II is totally positive if and only if

$$\begin{aligned} a_1+a_2+\cdots +a_n = 3n-6. \end{aligned}$$

Indeed, the total number of triangles in a triangulation is \(n-2\), and each triangle has three angles that contribute to a quiddity.

Remark 7.10

The above formula is equivalent to Theorem 1 of [14]. Moreover, it holds not only for the complete Farey sequence, but also for an arbitrary path in the Farey graph. Consider the Farey sequence of order 5 presented in Fig. 1. It has many different shorter paths, for instance, \(\left\{ \frac{1}{1},\frac{2}{3},\frac{3}{5},\frac{1}{2},\frac{1}{3},\frac{0}{1}\right\} \).

1.5 Totally positive solutions of Problem III

A solution \((a_1,\ldots ,a_n)\) of Problem III is totally positive if its double \((a_1,\ldots ,a_n,a_1,\ldots ,a_n)\) is a totally positive solution of Problem II. Every totally positive solution can be obtained from one of the solutions \((a_1,a_2)=(1,2)\), or (2, 1) by a sequence of operations (1.2).

The Conway–Coxeter theorem implies that there is a one-to-one correspondence between totally positive solutions of Problem III and centrally symmetric triangulations of 2n-gons; see also [6].

Example 7.11

There exist 70 different centrally symmetric triangulations of the decagon, for instance

The corresponding sequences \( (a_1,a_2,a_3,a_4,a_5)= (5,2,2,2,1), \, (4,3,1,3,1), \, (4,2,1,4,1),\ldots \) are totally positive solutions of Problem III.

The total number of totally positive solutions of Problem III is given by the central binomial coefficient \(\left( {\begin{array}{c}2n\\ n\end{array}}\right) =1, 2, 6, 20, 70, 252, 924,\ldots \)

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Ovsienko, V. Partitions of unity in \(\mathrm {SL}(2,\mathbb Z)\), negative continued fractions, and dissections of polygons. Res Math Sci 5, 21 (2018). https://doi.org/10.1007/s40687-018-0139-z

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