ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 19 (2020) 155-171 https://doi.org/10.26493/1855-3974.2286.ece (Also available at http://amc-journal.eu) Oriented area as a Morse function on polygon spaces* Daniil Mamaevf © Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29, Saint Petersburg 199178 Russia Received 20 March 2020, accepted 16 July 2020, published online 13 November 2020 We study polygon spaces arising from planar configurations of necklaces with some of the beads fixed and some of the beads sliding freely. These spaces include configuration spaces of flexible polygons and some other natural polygon spaces. We characterise critical points of the oriented area function in geometric terms and give a formula for the Morse indices. Thus we obtain a generalisation of isoperimetric theorems for polygons in the plane. Keywords: Flexible polygons, configuration spaces, Morse index, critical points. Math. Subj. Class. (2020): 58K05, 52B60 1 Preliminaries: necklaces, configuration spaces, and the oriented area function Suppose one has a closed string with a number of labelled beads, a necklace. Some of the beads are fixed and some can slide freely (although the beads never pass through one another). Having the necklace in hand, one can try to put it on the plane in such a way that the string is strained between every two consecutive beads. We will call this a (strained planar) configuration of the necklace. The space of all configurations (up to rotations and translations) of a given necklace, called the configuration space of the necklace, together with the oriented area function on it is the main object of the present paper. Let us now be precise. Given a tuple (n1,...,nk) of positive integers and a tuple (L1,...,Lk) of positive reals, we define a necklace N to be a tuple ((n1, L1),..., (nk, Lk)) interpreted as follows: *I am deeply indebted to Gaiane Panina for posing the problem and supervising my research. I am also thankful to Joseph Gordon and Alena Zhukova for fruitful discussions and to Nathan Blacher for his valuable comments on the linguistic quality of the paper. ÎThe research is supported by «Native towns», a social investment program of PJSC «Gazprom Neft». E-mail address: dan.mamaev@gmail.com (Daniil Mamaev) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 155 Ars Math. Contemp. 19 (2020) 125-145 • the necklace has the total of n = n(N) = n1 + • • • + nk beads on it; • k = k(N) of the beads are fixed and numbered by the index j = 1,..., k in counterclockwise order, the index j is considered to be cyclic (that is, j = 6k + 5 is the same as j = 5); • there are (nj - 1) freely sliding beads between the j-th and the (j + 1)-th fixed bead; • the total length of the string is L = L(N) = L1 + • • • + Lk; • the length of the string between the j -th and the (j + 1) -th fixed bead is equal to Lj. We fix some notation concerning polygons. • A planar n-gon is a collection of n (labelled) points (called vertices) (p1,... ,pn) in the Euclidean plane R2. Note that all kinds of degenerations, including self-intersection and collision of vertices, are allowed. • The space of all planar n-gons Polyn is thus just (R2)n. • The edges of a polygon P = (p1,..., pn) are the segments pipi+1 for i = 1,..., n, the length of the i-th edge is li = li (P) = |pjpi+11. Note that the index i = 1,..., n is cyclic (that is, i = 10n + 3 is the same as i = 3). To avoid messy indices, we introduce some additional notation associated with a necklace N = ((n1, L1),..., (nk, Lk)) (see Figure 1 for an example). For index j = 1,..., k • denote by j* the set of indices corresponding to the j-th piece of N: j* = {ni +-----+ nj_i + 1,. .., ni +-----+ nj}; (1.1) • define a function Lj : Polyn ^ R, the total length of the edges of a polygon corresponding to the j-th piece of N, that is Lj (p ) = E mp ). iej* Pi. „P7 ,P2 Pi p.5 = Pa ¿5 = 0 1* = {1,2}, * = {3}, 3* = {4,5,6,7}; s( 1) = 1, 8(2) = 3, 8(3) = 4; £i(P) = h + l2 = la; £ (P) = l3 = L ; £3 (P) = h + k + lr = L3. P3 Figure 1: A configuration P = (pi,.. ,,p7) of N = ((2,Li), (1,L2 ), (4, L3)). Definition 1.1. A (strained planar) configuration of a necklace N = ((n1, L1),..., (nk,Lk)) is a polygon P G Polyn with Lj(P) = Lj for all j = 1,...,k. D. Mamaev: Oriented area as a Morse function on polygon spaces 157 All configurations of a necklace N = ((ni, L1),..., (nk, Lk)) modulo translations and rotations form a space M(N) = M((n1, L1),..., (nk, Lk)) called configuration space of the necklace N. More formally, • Consider all the strained planar configurations of N = ((n1, L1),..., (nk, Lfc)) : M(N) = {P e Poly„ : Lj (P ) = Lj for j = l,...,k} . • The group Iso+ (R2) of orientation-preserving isometries of the Euclidean plane R2 acts diagonally on the space of all planar n-gons Polyn = (R2) n. • The configuration space of the necklace N is the space of orbits: M(N) = M(N)/lso+ (R2) . Definition 1.2. The oriented area A of an n-gon P = ((x1,y1),..., (xn, yn)) e (R2)n is defined to be )=2 X1 X2 yi V2 1 + 2 X2 X3 V2 V3 1 + ••• +2 xn X1 yn yi (1.2) The oriented area is preserved by the action of Iso+ (R2) and thus gives rise to a well-defined continuous function on M(N) for all the necklaces N. We will denote these functions by the same letter A. The study of critical points (i.e. the solutions of dA(P) = 0) of A: M(N) ^ R is the subject of the present paper. The paper is organised as follows. In Section 2 we review previously studied extreme cases: polygonal linkages (the 'all beads are fixed' case) and polygons with fixed perimeter (the 'one bead is fixed' case, which is clearly the same as 'none of the beads are fixed' case). In Section 3 we discuss the regularity properties of configuration spaces of necklaces. In the subsequent sections we study the non-singular part of the configuration space. In Section 4 we give a geometric description of critical points of the oriented area in the general case (Theorem 4.1) and deduce a formula for their Morse indices (Theorem 4.2). In Section 5 the auxiliary Lemmata 4.3 and 4.4 concerning orthogonality of certain spaces with respect to the Hessian form of the oriented area function are proven. In Section 6 we discuss the 'two consecutive beads are fixed' case and give a proof of Lemma 4.5. 2 An overview of existing results 2.1 Configuration spaces of polygonal linkages In the notation of the present paper these are the spaces M((1, /1),..., (1, ln)), i.e. the spaces M(N) for necklaces N with all beads being fixed. These spaces are studied in many aspects (see e.g. [1] or [2] for a thorough survey). On the side of studying the oriented area on these spaces, the first general fact about its critical points was noticed by Thomas Banchoff (unpublished), reproved by Khimshiashvili and Panina [5] (their technique required some non-degeneracy assumptions) and then proved again by Leger [8] in full generality. Theorem 2.1 (Critical configurations in the 'all beads are fixed' case; Bunchoff, Khimshiashvili, Leger, and Panina). Let N be a necklace with all the beads fixed. Then a polygon P € Msm(N) is a critical point of A if and only if it is cyclic (i.e. inscribed in a circle). 158 Ars Math. Contemp. 19 (2020) 125-145 After describing critical points, the following natural question arises: are these critical points Morse (i.e. whether HessP A, the Hessian of A at P, is a non-degenerate bilinear form on TPM(N)) and if they are, what is the Morse index (the maximal dimension of a subspace on which HessP A is negative definite). The state-of-the-art answer to this question requires some more notation (see Figure 2 for an example). Figure 2: Notation for a cyclic polygon. Definition 2.2. Let P be a cyclic n-gon, o be the circumcentre of P, and i € {1, • The central half-angle of the i-th edge of P is |Zpjopi+i| at(P) = 7+1' € [0,n/2]. • The orientation of the ¿-th edge of P is {1, if ZpiOpi+i € (0, n); 0, if ZpiOpi+1 €{0,n}; (2.1) -1, if ZpiOpi+1 € (-n, 0). We will denote by Cn the configuration space of cyclic n-gons with at least three vertices. More precisely, Cn={p € M- : AtisS,nS(P;lL8o,ni}/ «. (2-2) For P € Cn denote by Qp its circumscribed circle, by oP its circumcentre, and by RP the radius of QP. } n D. Mamaev: Oriented area as a Morse function on polygon spaces 159 Definition 2.3. Let P be a cyclic polygon with at least three distinct vertices. It is called admissible if no edge of P passes through its circumcentre o. In this case its winding number wP = w(P, o) with respect to o is well-defined. Theorem 2.4 (Morse indices in the 'all beads are fixed' case; Gordon, Khimshiashvili, Panina, Teplitskaya, and Zhukova). Let N = ((1, l1),..., (1, l„)) be a necklace without freely moving beads, and let P G Msm(N) be an admissible cyclic polygon. Then P is a Morse point of A if and only if "=1 ei tan ai =0 and in this case its Morse index is The formula more or less explicitly appeared in [6, 9], and [11], but in this form, with the precise condition of being Morse, the theorem was proved only in [3]. The following definition was first given in [3]. Definition 2.5. An admissible cyclic polygon P is called bifurcating if Yh=1 £i tan ai = 0. 2.2 Configuration space of n-gons with fixed perimeter This is the space M((n, L)) = M(n, L) (for different L these spaces are isomorphic, so usually L is set to 1). It is no secret since antiquity, that, with perimeter fixed, convex regular polygons maximise the area. All the critical points of the oriented area together with their indices were determined only in a recent paper [7] by Khimshiashvili, Panina and Siersma. Definition 2.6. A regular star is a cyclic polygon P such that all its edges are equal and have the same orientation (see (2.1)). A complete fold is a regular star P with pi = pi+2 for all i = 1,..., n. It exists for even n only. Theorem 2.7 (Critical configurations and Morse indices in the 'one bead is fixed' case; Khimshiashvili, Panina, and Siersma). (1) M(n,L) is homeomorphic to CPn-2. (2) A polygon P G Msm(n, L) is a critical point of A if and only if it is a regular star. (3) The stars with maximal winding numbers are Morse critical points of A. (4) Under assumption that all regular stars are Morse critical points, the Morse indices Remark 2.8. The super-index in mP (A) allows one to identify the domain of A. For example, mP(A) is the Morse index of A: M (n, L1(P)) ^ R at point P (as in (4) of Theorem 2.7), and Mp'"'1 (A) is the Morse index of A: M))1, Li(P)0,..., )1, L„(P)00 ^ R at point P (as in Theorem 2.4). This notation will be of much use in the proof of Theorem 4.2. Mp(A) = #{i G {1, .. ., n} : £i > 0} - 1 - 2wp 0, if Ya=1 £i tan ai > 0; 1, otherwise. are: if wp < 0; MP (A) = D- 2, if wP > 0; if P is a complete fold. 160 Ars Math. Contemp. 19 (2020) 125-145 Let us also mention an auxiliary statement proven in [7]. Lemma 2.9 (Khimshiashvili, Panina, and Siersma). Let P be a regular star which is not a complete fold with wP > 0. Then P is a non-degenerate local maximum on Cn. In fact, this lemma together with Theorem 2.4 and Lemma 4.3 allows one to omit the assumption in (4) of Theorem 2.7. All the critical points of the oriented area on Cn were described in a recent preprint [10] by Siersma. 3 Singular locus of the configuration space Definition 3.1. Let P be a configuration of a necklace N = ((ni, L1),..., (nk, Lfc)). It is called non-singular if L = (L1,..., Lk) is a smooth submersion at P (i.e. L is differ- entiate at P and its differential DP L : TP Polyn otherwise it is called singular. ^ TPRk is a surjective linear map), First we give a geometric characterisation of singular configurations. Consider a polygon P = (pi,... ,p„), with Pi = (xi,yi) G R2 and k = |pi+i - pi| = 0 for all i = 1,... ,n. Define fa to be the oriented angle between vectors (1,0) and (pi+i -pi). Denote by s(j) = ni + • • • + nj_i + 1 the index of the j-th fixed bead. Then every Lj is differentiable at P and the derivative of Lj with respect to xi and yi is as follows: — cos Pi if i = s(j); ë (p )= dj (p ) = dyi cos Pi-1 — cos Pi, if i G j* \ {s(j)}; cos Pi-1, — sin pi, if i = s(j + 1); otherwise. if i = s(j); sin Pi-i — sin Pi, if i G j * \ {s(j)}; sin Pi-1, if i = s(j + 1); . 0, otherwise. (3.1) (3.2) Definition 3.2. Let N = ((«i, ¿1),..., ,Lk)) be a necklace. An index i G {1,..., n(N)} is called boundary if it is equal to s(j) for some j G {1,..., k}, otherwise it is called inner. In Figure 1 the indices 1,3,4 are boundary and the indices 2, 5, 6,7 are inner. Lemma 3.3. A configuration P G Polyn of the necklace N = ((n1, L1),..., (nk ,Lk)) is singular if and only if one of the following holds: (1) li = 0 for some i G {1,..., n(N)}; (2) P fits in a straight line in such a way that = ¡3i-1 for all inner indices i. Proof. The first condition is equivalent to L being differentiable at P. Therefore, what is left to prove, is that for P g M(N) with no vanishing edges, the second condition holds if and only if the gradients gradP L1,..., gradP Lk are linearly dependent. Suppose that A1 gradP L1 + • • • + Xk gradP Lk =0 is a non-trivial vanishing linear combination. If A j = 0, then, using formulae (3.1) and (3.2) for boundary index s(j), D. Mamaev: Oriented area as a Morse function on polygon spaces 161 we get —Aj cos+ A¿_i cos = 0 and — Aj sin+ A¿_i sin= 0. It means that points A j (cos ,0s(j), sin ,0s(j)) = (0,0) and Aj-1(cos ^s(j-)_1, sin ^s(j)-1) coincide, which implies that 2(,0s(j) — Ps(j)_i) =0 and Aj_1 = cos(,0s(j) — Ps(j)_1)Aj = 0. It follows then that A j = 0 for all j = 1,..., k, consequently, (we now use (3.1) and (3.2) for inner indices) $ = for all inner indices i, meaning that P is composed of straight segments of lengths L1,...,Lk. Taking into account previously deduced formula 2(fti — Pi_\) = 0 for boundary i, we conclude that P does satisfy condition (2). Reversing the above argument, we get the reverse implication. □ Definition 3.4. A singular configuration P of a necklace N is called strongly singular, if it satisfies (2) in Lemma 3.3. Otherwise it is called weakly singular. Remark 3.5. Let N = ((n1, L1),..., (nk,Lk)) be a necklace. Then • weakly singular configurations of N are, in a sense, inessential (for instance, M(N) is a topological manifold around them); • if (L1,..., Ln) is such that ±L1 ± • • • ± Ln = 0 for any choice of ± (such tuples are called generic in [2]), then there are no strongly singular configurations of N. Together, these two facts allow to deduce some information about topology of M(N) for generic N from Theorems 4.1 and 4.2, but this is not the subject of the present paper. Now let Msm(N) be the set of non-singular configurations of necklace N and Msm(N) be the non-singular part of M(N): Msm(N)=Mw)={p - poiy.: p0isfi™;-on,sli;;}/i»+ ^ If these spaces are non-empty, they are smooth manifolds. This statement generalises previous results on smoothness of configuration spaces of polygonal linkages by Kapovich-Millson [4] and Farber [2]. Definition 3.6. A necklace N = ((n1, L1),..., (nk, Lk)) is called realisable if for all j = 1,..., k, such that nj = 1, the inequality 2Lj < L1 + • • • + Lk holds. Proposition 3.7. Let N be a realisable necklace. Then (1) Msm(N) is a smooth (2n — k)-dimensional submanifold of Polyn = R2n; (2) Msm(N) is a topological manifold of dimension 2n — k — 3 with a unique smooth structure making the quotient map Msm(N) ^ Msm(N) a smooth submersion; (3) the oriented area function A is a smooth function on Msm(N). Proof. It follows from Lemma 3.3, that the inequalities 2Lj < L1 + • • • + Lk are necessary and sufficient for Msm(N) to be non-empty. The first claim is clear since Msm(N) is locally a level of a smooth submersion L = (£1,..., Lk): (R2)" ^ Rk. To establish the second claim, we first note that Msm(N) is an orbit space of the action of 3-dimensional Lie group Iso+ (R2) on the smooth manifold Msm(N). Thus, it suffices to observe that the action is free and proper, which is indeed the case. The third claim is obvious since the smooth structure on Msm(N) is induced from Polyn, and the oriented area A is a smooth function (cf. (1.2)) on Polyn preserved by the action of Iso+ (R2). □ 162 Ars Math. Contemp. 19 (2020) 125-145 4 Main results: critical configurations and their Morse indices in the general case The following theorem describes critical points of the oriented area on configuration spaces of necklaces. It generalises Theorem 2.1 and (2) in Theorem 2.7. Theorem 4.1 (Critical configurations in the general case). A polygon P G Msm((«, Li),..., (nk, Lk)) is a critical point of A if and only if all of the following conditions hold: (1) P is cyclic; (2) li(P) = Lj/« for all i G j*; (3) eh (P) = e¿2 (P) for all ii,Í2 G j*, where j * is the set of indices corresponding to the j-th piece of a necklace (see (1.1)) and £j(P) is the orientation of the i-th edge of a cyclic polygon P (see (2.1)). The proof essentially is a reformulation of geometric arguments into the language of Lagrange multipliers, so we first write partial derivatives of A with respect to xi and y¿: 3A 2 • ^ (P) = li-i sin A-i + li sin A (4.1) 3A 2 • 7— (P) = -li-1 cos pi-1 — li cos pi. (4.2) dyi We follow the convention 0 • undefined = 0 hence both sides are defined for all P G Polyn. Proof of Theorem 4.1. Let P be a non-singular configuration of a necklace N = ((n1, L1),..., (nk, Lk)). Then P is a critical point of A if and only if there exist A1,..., Ak G R, such that 2 • gradP A = A1 gradP L1 + • • • + Ak gradP Lk. Assume that 2 • gradP A = A1 gradP L1 + • • • + Ak gradP Lk. Applying formulae (3.1), (3.2), (4.1), (4.2) to an inner index i G j*, one deduces li-1 sin Pi-1 + li sin Pi = Aj (cos Pi-1 - cos Pi); —li-1 cospi-1 — li cospi = Aj (sinpi-1 — sinpi). If pi = pi-1, then li = li-1 = 0, but P is non-singular, so it cannot be the case by Lemma 4.3. The only other possibility for these equations to hold is li-1 = li and Aj = li cot (ft-*-1). Since we have such equations for all inner indices corresponding to j, we get li1 = li2 for all i1, i2 G j *, which implies condition (2) of the theorem. Moreover, for all i G j * \ s(j) we get cot (P*-^-1} = íí^í , therefore pi — pi-1 is the same for all i G j* \ s(j), which implies that there is a circle Qj with centre oj such that conditions (2) and (3) of the theorem hold. It now remains to prove that P is cyclic, i.e. oj is the same for all j = 1,..., k. If P is a smooth point of Msm((1, l1),..., (1, ln)) C Msm{N), in other words, if P does not fit in a straight line, then we are done by Theorem 2.1. Suppose that P fits in a straight line. Pick a boundary vertex i = s(j + 1), denote lj = Lj /«j, and apply formulae (3.1), (3.2), (4.1), (4.2) to i: lj sin pi-1 + lj+1 sin pi = Aj cos pi-1 — Aj+1 cos pi; —lj cos pi-1 — lj+1 cos pi = Aj sin pi-1 — Aj+1 sin pi. D. Mamaev: Oriented area as a Morse function on polygon spaces 163 Since P fits in a straight line, 2(ft.¿ - ft.¿_1) = 0. If ft^ = ft, = ft, then the points (lj + lj+1)(cos ft, sin ft) and (\¿ - Aj+1)(cos(ft + n/2), sin(ft + n/2)) coincide which cannot be the case since lj, lj+1 > 0. If fti_1 = ft, + n = ft + n, then the points (lj - lj+1)(cos ft, sin ft) and (A¿ + Aj+1)(cos(ft + n/2), sin(ft + n/2)) coincide, which implies that lj = lj+1. Since this is the case for all j, P is a complete fold and thus indeed is cyclic. Now assume that a non-singular configuration P of necklace N satisfies conditions (1)-(3). Let Q be its circumscribed circle with centre o. Denote by Yj the oriented angle Zps(j)ops(j)+1 and set A¿ = l, cot (y¿/2) for some index i corresponding to j. Since y¿ = ft, - fti_1 for inner indices i, equality 2 • gradP A = A1 gradP L1 + • • • + Ak gradP Lk holds in all inner indices. For a boundary index i = s(j + 1) we can (performing rotation around o) assume that fti_1 = 0, and what we need to check then is the following two equalities: lj+1 sin ft, = lj cot(7i_1/2) - lj+1 cot(7i/2) cos ft,; -lj - lj+1 cos ft, = -lj+1 cot(7,/2) sin ft,, Putting the origin at o, we note that Pi+1 - Pi = lj+1 • (cos fti, sin fti), Pi+1 + Pi = lj+1 cot Y;^ • (-sin ft,, cos ft,), p = ( jcot ^ and thus the desired equalities are just the coordinate manifestations of the obvious identity Pi+1 - Pi Pi+1 + Pi 2 2 + Pi = (0,0). □ The following theorem provides a criterion for an admissible cyclic polygon to be a Morse point of the oriented area and gives a formula for its Morse index. It generalises Theorem 2.4 and allows one to omit the assumption in (4) of Theorem 2.7. Theorem 4.2 (Morse indices in the general case). Let N = ((ni, Li),..., (nk, Lk)) be a realisable necklace (see Definition 3.6), and P G Msm(N) be an admissible (see Definition 2.3) critical point of the oriented area A Then P is a Morse point of A if and only if it is not a bifurcating polygon (see Definition 2.5). In this case its Morse index can be computed as follows: 1 A, f0, ifY]k -, n, E, tan A, > 0; "p(A)=1 s<2n, -1) • +1) -1 -2wp -1, new*' ' ' ; where E, = e, and A, = a, for some i G j* (due to Theorem 4.1 this does not depend on the choice of i). Proof. Let P be as in the theorem. First, let us split the tangent space of Msm(N) at the critical point P into subspaces that are orthogonal with respect to the Hessian form HessP A. For this, given a polygon P, we introduce the following submanifolds in Msm(N): 164 Ars Math. Contemp. 19 (2020) 125-145 (1) EP = Msm((1, l1),..., (n, ln)) C Msm(N) is the space of all polygons having the same edge length as P; (2) CP = Msm(N) n C is the subspace of cyclic polygons; , (qs(j),.. .,qs(j+1)) is cyclic] qi = Pi for i £ j* \ {s(j)} (3) Cp = Q e Msm(N) : v \ for j = 1,...,k. We will deduce the theorem from Lemmata 4.3, 4.4, and 4.5 (see Sections 5 and 6 for their proofs). Lemma 4.3. Let P be as in Theorem 4.2. Then (1) EP C Msm(N) is a smooth (n — 3)-dimensional submanifold in a neighbourhood of P ; (2) CP C Msm(N) is a smooth (n — k)-dimensional submanifold in a neighbourhood of P; (3) EP and CP intersect transversally at P, i.e. TPMsm(N) = TPEP 0 TPCP; (4) TP EP and TP CP are orthogonal with respect to the bilinear form HessP A. One can note that none of the Cp are contained in CP. Nonetheless, from the following lemma one sees that in the first approximation they very much are. Lemma 4.4. Let P be as in Theorem 4.2. Then (1) Cp C Msm(N) is a smooth (nj — I)-dimensional submanifold in a neighbourhood of P; (2) Tp C P = 0 TP Cp f=i (3) TP Cp are pairwise orthogonal with respect to the bilinear form HessP A. It remains to compute the Morse index of P with respect to A on each Cp. Lemma 4.5. Suppose that P e Cn+i is such that h = • • • = ln = L/n and £i = 1 (£i = —1) for i = 1,... ,n. Then P is a non-degenerate local maximum (minimum) of the oriented area on Msm((n, L), (1, ln)) n Cn+1. Now we are ready to prove the theorem. From Lemmata 4.3 and 4.4, P is a Morse point of A on Msm if and only if it is a Morse point of A on Ep and all of Cp. Since P is always a Morse point on each Cp (because by Lemma 4.5 it is a non-degenerate local extremum), it is a Morse point of A on Msm if and only if it is a Morse point of A on Ep, which is equivalent to P not being bifurcating by Theorem 2.4. Moreover, Lemmata 4.3 and 4.4 imply that if P is a Morse point of A on Msm, then its Morse index is k P (A) = npP(A) + nCpP(A) = up,-'1 (A) + ^up (A). f=i D. Mamaev: Oriented area as a Morse function on polygon spaces 165 From Theorem 2.4 we know that iE+D -1 - - - {;■ 3Ej tan A >0; 3=1 y From Lemma 4.5 and (1) of Lemma 4.4 we get cp 1 MP (A) = -(uj - 1) • (Ej + 1). Summing all up, we obtain the desired formula. □ 5 Orthogonality with respect to the Hessian form of the oriented area Let us remind that Cn is the configuration space of cyclic polygons with at least three different vertices (see (2.2)). First, we parametrise Cn smoothly. For this we introduce Hn = {(01,...,0n) e (S1)" :#{01,...,M > 3}/S\ where S1 acts on (S1)" diagonally by rotations. Consider the following map p: (^S1 )n \ Diag^ x R>0 ^ (R2)" \ Diag, (61,... ,8n, R) ^ R • ((cos 91, sin 81),..., (cos 9n, sin 8n)). Lemma 5.1. p induces a diffeomorphism p: Hn x R>0 ^ Cn. Proof. p is obviously a bijection, so the only thing we need to check is that the Jacobian of p has rank (u + 1) at every point. In fact, it is just a statement of the form 'S1 x R>0 is diffeomorphic to R2 \ {0} via polar coordinates', but we compute the Jacobian for the sake of completeness: / Jac1 p \ f-Rsin81 Rcos81 0 ... 0 0 \ Jac p = Jacn p \Jacn+1 pj R sin 8n R cos 8 n cos 81 sin 81 cos 82 . . . cos 8n sin 8n The first u rows are obviously linearly independent. Suppose one has Jacn+1 p = A1 Jac1 p + • • • + An Jacn p. Then for any i = 1,..., u one gets (cos 8j, sin 8j) = A1(-Rsin 8h Rcos 8,) = A1R (cos (8, + n/2), cos (8, + n/2)), which implies A, = 0, a contradiction. □ We now provide local coordinates for Cn. Lemma 5.2. Let P e Cn be an admissible non-bifurcating cyclic polygon with edge lengths l1,..., ln > 0. For Q e Cn let t,(Q) = (Q) - l,. Then (t1,... ,tn) are smooth local coordinates for Cn around P. 166 Ars Math. Contemp. 19 (2020) 125-145 Proof. In view of Lemma 5.1 we just need to show that for 0: Hn x R>o ^ R— (01,... ,0—,R) ^ R ■ (y2 - 2cos(02 - 01),..., V 2 - 2cos(^ - 0—)) Jac 0 is of rank n at points where 01 = 02 = ! R sin(gi —) 0 ... ^2-2cos(0i-02) R sin(02— gi) Rsin(02—^3) ^2 — 2cos(02 — gi) ^2 —2cos(02 —®3) = 0n = . Indeed, Jac 0 is R sin(gi— gn) \ ^2 —2cos(gi—gn) 0 R sin( -gn) -y/2 —2 cos(gn-i—gn) R sin(gn — gn-i) ^2 — 2cos(gn— gn-i) a/2 —2 c R sin( -gi) "gi ) \^2 —2cos(g2 —gi) ^2 —2cos(g3 —g2) ^2 —2cos(gn—gn-i) ^2 —2cos(gi—gn ) / Since 2(0i+1 - 6i) = 0, all the entries are well-defined and non-zero. Consider a vanishing non-trivial linear combination of columns. The form of the first n rows forces the coefficient at the ¿-th column to be equal (up to the common multiplier) to ^^(g^—g^f)^, but then for the last row we have E2 — 2 cos(0i — 0i+1) v—^ ( 0i — 0i+1 -. i„—^ = i i+1 2 which means exactly that P is bifurcating and contradicts the assumptions of the lemma. Thus, Jac 0 has rank n as desired. □ Proof of Lemma 4.3. To prove the first two claims let us note that smooth structures on EP, CP, and Msm(N) come from the smooth structure on Poly— = (R2)—. Thus, the first claim immediately follows from Lemma 3.3, as the only cyclic polygon fitting into a straight line is a complete fold, which is not admissible. The dimension of EP is computed according to (2) in Proposition 3.7. From Lemma 5.1 it follows that C— around P is a smooth submanifold in Poly— / Iso+, and from Lemma 5.2 we deduce that CP around P is a smooth (n - k)-dimensional submanifold of C— as it is a preimage of the linear subspace of codimension k in R— under the map Q ^ (t1(Q),... ,t—(Q)). Thus the second claim is also proved. The third claim is equivalent (by dimension count) to representability of every vector in TPMsm (N) as a sum of two vectors from TPEP and TPCP respectively, but this is indeed the case since every polygon Q near P in Msm(N) can be obtained first by a move in CP making the edges of desired length (by Lemma 5.2) and then by a move inside EQ. 0 0 ni 0 0 0 0 0 D. Mamaev: Oriented area as a Morse function on polygon spaces 167 Finally, we establish the fourth claim. Consider v e Tp C and w e Tp E. To compute Hessp A(v, w), we choose a curve 7: (-e, e) ^ TPC such that 7(0) = P and y'(0) = v, then we extend w to a vector field W(t) e T7(t)E7(t) along 7. Then HessP*4 (v,w) = — P v ' dt d7(t)A(W (t)). t=0 But dY(t)A vanishes on T7(t)E7(t) by Theorem 4.1. □ The following lemma allows one to relate Cp with Cp. Lemma 5.3. Let P e Cn be an admissible non-bifurcating cyclic polygon such that li = l2 and Zpiop2 = Zp2op3, where o is the centre of the circumscribed circle Q. Let V be a local vector field around P equal to (¿|--dr) in the coordinates from Lemma 5.2. Then (VR) (P) = 0 and (Vdo6) (P) = 0for a,2& e {1,... ,n} \ {2}, where Vf is the derivative of a function f along V, R(Q) is the radius of the circumscribed circle of Q and dab(Q) = |qb - qa|. Proof. Consider a curve P (s): (—e, e) ^ Cn, (ti ,...,tn)(P (s)) = (s,-s, 0,..., 0). We choose representatives P(s) e Polyn in such a way that Op(s) = (0,0) and (p3 - pi) is codirectional with x-axes. Notice that -P(-s) is obtained from P(s) by the following procedure: Pi(-s) = p^s) for i = 2 andp2(-s) is symmetric to p2(s) relative to y-axes. From this it follows that P(s) — P(- s)) = (0,0,2n, 0,..., 0) for some n > 0. Hence all Pi for i = 2 are not moving in the first approximation, which implies the statement of the lemma. □ Proof of Lemma 4.4. The space {Q e Msm(N) : qi = pi for i e j * \{s(j)}} is a smooth submanifoldin Msm(N) diffeomorphic to Msm((nj,Lj), (1, |ps(j+i) -ps(j)|)). Under this identification, Cp is just Cp. Applying (2) of Lemma 4.3 to Msm((nj, Lj), (1, |ps(j+i) - ps(j)|)), we get the first claim. To establish the second claim it suffices to find bases in every TPCp such that their disjoint union forms a basis of TPCp. Consider the coordinates from Lemma 5.2. On the one hand, when we consider cyclic polygons coordinatised by (ti,..., tn), the vectors (dt8 - d^) for inner i form a basis of TPCp. On the other hand, when we consider Cp coordinatised by (si)i£j.\{s(j)}, where si = li(Q) - li(P), the vectors (q^ - d^) for i e j* \ {s(j)} formabasisof TPCp. It now follows from Lemma 5.3 that (dtd - -J^) = (dsd t - ¿d^) for i e j* \ {s(j)} thus the second claim is proven. Now we pass to the third claim. Consider v e TpCp and w e TpCp, take a curve Y: (-e, e) ^ Cp such that 7(0) = P and y'(0) = v, and a curve a: (-e, e) ^ Cp, such that a(0) = P and a'(0) = w. Then extend w to a vector field W(t) e T7(t)Msm(N) along y by setting W(t) = a' (0), where at: (-e, e) ^ C^(t) is such that at(0) = 7(t) and for all i e j * \ s(j) the i-th vertex of at(s) is the same as the i - th vertex of a(s). Then Hessp A(v,w)= d W (t)A, dt t=0 and it vanishes since W (t)A does not depend on t. □ 168 Ars Math. Contemp. 19 (2020) 125-145 6 Configuration spaces of polygons with perimeter and one edge length fixed These are the spaces M ((n, L), (1, Z)) for L > Z. Vividly speaking, it is the space of broken lines of given length with fixed endpoints. One can think that the first and the last vertices have coordinates (0,0) and (Z, 0) respectively. Our interest in these spaces was first motivated by the fact that they are simple enough to be studied completely, but then it turned out that they are important for understanding the case of general necklaces. Proposition 6.1 (Configuration space in the 'two consecutive beads are fixed' case). Let L > Z and n > 2. Then M ((n, L), (1, Z)) is homeomorphic to the sphere S2n-3. Proof. By setting p1 = (0,0) and pn+1 = (Z, 0) we identify M ((n, L), (1, Z)) with the level set F-1(L) = {(P2,... ,pn) e (R2)n-i : F(P2, . . . ,pn) = l} , where F (p2, . . . ,Pn) = |p2 | + |p3 - P2 | +-----+ |pn - Pn-i| + |(Z, 0) - Pn|. F is a convex function as sum of convex functions. The sublevel set F-1((-to, L]) is bounded since if any of |p41 is greater than L, then F (p2,..., pn) > L by triangle inequality. Also, the set F-1((-to, L)) is non-empty, since if all of the pi are in the disk of radius 6 around (Z/2,0), then F (p2,...,pn) < (z/2 + 6) + (n - 3)6 +(Z/2 + 6) = Z + (n - 1)6, which is less than L for small 6. So, F-1(L) is a boundary of the compact convex set F-1((-to,L]) c (R2)n 1 with non-empty interior and thus is homeomorphic to S2n-3. □ The following two propositions are easily deduced from Theorems 4.1 and 4.2 respectively. Proposition 6.2 (Critical points in the 'two consecutive beads are fixed' case). Let L > Z and n > 2. Then critical points of A on Msm((n, L), (1, Z)) are in bijection with the solutions of nZ |Un-1(x)| = - (6.1) where Un-1 is the (n - 1)-th Chebyshev polynomial of second kind, that is, Un- 1(cos = ^^. n 1 ^ ~ > sin ^ Proof. By Theorem 4.1 a configuration P e Msm((n, L), (1, Z)) is a critical point of A if and only if it is inscribed in a circle Q with centre o and radius R in such a way that Zp1op2 = • • • = Zpnopn+1 =: a(P) = a. Let us construct a bijection {critical points of A on Msm((n, L), (1, Z))} ^ {solutions of (6.1)}, P ^ cP. Let cP = cos(a/2), where a/2 e (0, n). Since L/n = 2 - 2 cos a = 2Rsin(a/2) and Z = ^^2 - 2cos(na) = 2R| sin(na/2)|, we get Un - 1(cP) = nZ/L. Since the map R ^^ \{0}^ (-1,1), a ^ cos(a/2) is a bijection and P is uniquely determined by a(P), the constructed map P ^ cP is indeed a bijection. □ D. Mamaev: Oriented area as a Morse function on polygon spaces 169 Proposition 6.3 (Morse index in the 'two consecutive beads are fixed' case). If P is an admissible non-bifurcating critical configuration of A on Msm((n, L), (1, l)), then its Morse index is „!.., I 2n — 2 — i, if cP is the i-th largest positive solution of (6.1); Mp' (A) = < I i — 1, if cP is the i-th smallest negative solution of (6.1). Proof. By symmetry reasons, to prove the claim, it suffices to prove it only for P with cP > 0. Then by Theorem 4.2 one has n,i, ,r, N , 1t , n fo, if ntan(a/2) > e„+i tan(na/2); MP' (A) = (2n — 1)^(£n+i + 1) — 1 — wP — < . P 2 1, otherwise. The roots and extrema of Un_ 1 (t) are interchanging. Let us start from t = 1 and move to the right. The extrema correspond to the bifurcating polygons (i.e. those with n tan(a/2) = e„+1 tan(na/2)) and the roots correspond to polygons with ln+1 = 0. So, when t passes a root, en+1 changes from 1 to —1 and whenever t passes an extrema, the last summand changes from 0 to 1. When p1pn +1 passes through o, wP increases by 1, and en+1 changes from — 1 to 1, which does not change the Morse index. The right-most t corresponds to the global maximum, so the above argument completes the prove. □ Finally, we check the last yet unproven ingredient in the proof of Theorem 4.2. Proof of Lemma 4.5. Let P be as in the lemma. Without loss of generality we can assume that QP = Q is the unit circle with centre o, and, due to symmetry, it is enough to prove the statement for P with wP > 0. We should prove that the function A • (polygons P inscribed in the unit circle with L(p) = L1 ^ R in+1 I ln+1(P) l attains a non-degenerate local maximum at P. For this it suffice to prove that the function G. ipolygons inscribed! r ' 1 in the unit circle | , 2A(Q) / L(Q)2 L^ ( L1(Q)2 L2 G(Q) = 7-77772 — A 1-77^2 — 12 — M ln+1(Q)2 Vln+1(Q)2 IV V ln+1 (Q)2 l2 (6.2) attains a non-degenerate local maximum at P for suitable A and m. Set a = Zp1op2 = • • • = Zpnopn+1 G (0, n) and introduce local coordinates by setting tj(Q) = Zqioqi+1 — a for i = 1,..., n. First, we write the functions involved in the definition (6.2) in these coordinates: ln+1 (t1,... ,tn) = ^ 2 — 2cos ^na + ^tjj ; I L1 (t1,..., tn) = V2 — 2cos(a +tj); i=1 nn 2A (t1,..., tn) = ^^ sin(a + tj) — sin I na + ^^ tj I . (a + tj) — sin na - =1 V i=1 170 Ars Math. Contemp. 19 (2020) 125-145 Secondly, we perform the computations in the 2-jets at the point P, which by the aforementioned coordinates are identified with R[t i,..., t„]/I, where I is the ideal generated by all products tjtjth with i, j, h = 1,..., n. It turns out that the 2-jets of the functions we are interested in are all contained in the subring R+RTi + RT2 + RI2, where Ti = ^"=1 tj and I2 = E"=112. This subring is naturally identified with the ring R[Ti, IWd3, T22, TiI2). With all the identifications done, the 2-jets of the functions involved in the definition (6.2) look as follows: J2U ■ = 1 • (i + 2 cot (na) Ti - 8i2) ; j2Li = i (l + 2n COt (2) T - sin a sin(na) 2 j2(2A) = (nsin a — sin(na)) + (cos a — cosna)!---—I2 +----I2. Now, setting x = tan a and y = tan nf, we can write the 2-jets of the summands in (6.2) in more or less compact form: ■ ( L L2) nx(1+ y2)(y — nx)T nx2(1 + y2)T + C ( 2. j2 ^ 12+i — 72 J = -y3(1+ x2)-Ti — 4y2(1 + x2) T2 + Ci(n, x, y)Tl. L2 L2 )2 _ n2x2(1 + y2)2(y — nx)2^. j2 72---- -„6/1 , - £+1 IV y6(i + x2 1 ; . / 2A 2A(P)\ (1 + y2)(y — nx) x(1 + y2) + C ( 2 H — —) = 2y3(1+ x2) Ti — 4y2(1 + x2)T + C2(n,x,y)TL To get rid of I\ in j2G we set A = ^, and then finally obtain j'2 0. As for the second one, nx — y = 0 as P is not bifurcating, and thus, whatever Ci and C2 are, when ^ is big enough the second term is a non-positive definite quadratic form. Therefore, G attains a non-degenerate local maximum at P for A = ^ and large positive □ ORCID iD Daniil Mamaev © https://orcid.org/0000-0002-7606-4276 References [1] R. Connelly and E. D. Demaine, Geometry and topology of polygonal linkages, in: C. D. Toth, J. O'Rourke and J. E. Goodman (eds.), Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, Florida, pp. 233-256, 2017, https://www.taylorfrancis. com/books/e/97 81315119601/chapters/10.1201/9781315119601-9. [2] M. Farber, Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008, doi:10.4171/054. D. Mamaev: Oriented area as a Morse function on polygon spaces 171 [3] J. Gordon, G. Panina and Y. Teplitskaya, Polygons with prescribed edge slopes: configuration space and extremal points of perimeter, Beitr. Algebra Geom. 60 (2019), 1-15, doi:10.1007/ s13366-018-0409-3. [4] M. Kapovich and J. Millson, On the moduli space of polygons in the euclidean plane, J. Differential Geom. 42 (1995), 430-464, doi:10.4310/jdg/1214457034. [5] G. Khimshiashvili and G. Panina, Cyclic polygons are critical points of area, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 360 (2008), 238-245, doi:10.1007/ s10958-009-9417-z. [6] G. Khimshiashvili and G. Panina, On the area of a polygonal linkage, Dokl. Akad. Nauk. Math. 85 (2012), 120-121, doi:10.1134/s1064562412010401. [7] G. Khimshiashvili, G. Panina and D. Siersma, Extremal areas of polygons with fixed perimeter, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 481 (2019), 136-145, doi: 10.1007/s10958-020-04835-9. [8] J. C. Leger, Aire, périmètre et polygones cocycliques, 2018, arXiv:1805.05423 [math.CO] . [9] G. Panina and A. Zhukova, Morse index of a cyclic polygon, Cent. Eur. J. Math. 9 (2011), 364-377, doi:10.2478/s11533-011-0011-5. [10] D. Siersma, Extremal areas of polygons, sliding along a circle, 2020, arXiv:2001.10882 [math.CO] . [11] A. Zhukova, Morse index of a cyclic polygon II, St. Petersburg Math. J. 24 (2013), 461-474, doi:10.1090/s1061-0022-2013-01247-7.