Closed planar curves without inflections

We define a computable topological invariant $\mu(\gamma)$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $\mu(\gamma)$ on a given $\gamma$.


Introduction.
In this paper, curves are always assumed to be regular (i.e. immersed). The well-known Fabricius-Bjerre [3] theorem asserts (see also [5]) that (0.1) d 1 (γ) − d 2 (γ) = # γ + i γ 2 holds for closed curves γ satisfying suitable genericity assumptions, where d 1 (γ) (resp. d 2 (γ)) is the number of double tangents of same side (resp. opposite side) and # γ and i γ are the number of crossings and the number of inflections on γ, respectively. affirmative answer to the Halpern conjecture in [6], the second author [8] proved the inequality for closed curves without inflections (see Remark 3.4). It is then natural to expect that there might be further obstructions for the topology of closed planar curves without inflections. In this paper, we define a computable topological invariant µ(γ) for closed planar curves. By applying the Gauss-Bonnet formula, we show the inequality i γ ≥ µ(γ), which is sharp at least for closed curves satisfying # γ ≤ 4. In fact, for such γ, there exists a closed curve σ which has the same topological type as γ such that i σ = µ(σ). As an application, we classify the topological types of closed planar curves satisfying i γ = 0 and # γ ≤ 5. Moreover, we discuss the relationship between the number of double tangents of the curve and the invariant I(γ) on a given γ.

Preliminaries and main results
We denote by R 2 the affine plane, and by S 2 the unit sphere in R 3 . A closed curve γ in R 2 or S 2 is called generic if (1) all crossings are transversal, and (2) the zeroes of curvature are nondegenerate. By stereographic projection, we can recognize S 2 = R 2 ∪ {∞}. Two generic closed curves γ 1 and γ 2 in R 2 (in S 2 ) are called geotopic, or said to have the same topological type, in R 2 (resp. in S 2 ) if there is an orientation preserving diffeomorphism ϕ of R 2 (resp. on S 2 ) such that Im γ 2 = ϕ(Im γ 1 ). This induces an equivalence relation on the set of closed curves. We denote equivalence classes by γ 1 (resp. [γ 1 ]). We fix a closed regular curve γ : S 1 → R 2 . A point c ∈ S 1 is called an inflection point of γ if det(γ(t),γ(t)) vanishes at t = c. We denote by i γ the number of inflection points on γ. A closed curve γ : [0, 1] → R 2 is called locally convex if i γ = 0. Whitney [13] proved that any two closed curves are regularly homotopic if and only if their rotation indices coincide. So then one can ask if this regular homotopy preserves the locally convexity when the given two curves are both locally convex. In fact, one can easily prove this by a modification of Whitney's argument, which has been pointed out in [9,p35,Exercise 11]. For the sake of the readers' convenience we shall outline the proof: Proposition 1.1. Let γ 1 and γ 2 be two locally convex closed regular curves. Suppose that γ 1 has the same rotation index as γ 2 . Then there exists a family of closed Proof. Suppose that the curves γ j (j = 1, 2) are both positively curved and have the same rotation indices, equal to m. We change, if necessary, the parametrizations of γ j so that the tangent vectorsγ j (t) are positive scalar multiples of cos t sin t . Define a homotopy Γ ε between γ 1 and γ 2 by It is easy to verify that {Γ ε } ε∈[0,1] satisfies the required conditions (1) and (2).
For a given generic closed curve γ, we set Since i γ is an even number, so is I(γ). Inflection points on curves in R 2 correspond to singular points on their Gauss maps. So it is natural to ask about the existence of topological restrictions on closed curves without inflections, in other words, we are interested in the topological type of closed curves γ satisfying I(γ) = 0. There are explicit combinatorial procedures for determining generic closed spherical curves with a given number of crossings, as in Carter [4] and Cairns and Elton [2] (see also Arnold [1]). We denote by # γ the number of crossings for a given generic closed curve. In this paper, we use the table of closed spherical curves with # γ ≤ 5 given in the appendix of [7].
For example, the table of closed spherical curves with # γ ≤ 2 is given in Figure  2, where 1 2 (resp. 2 2 ) means that the corresponding curve has 2-crossings and appears in the table of curves in [7] with # γ = 2 primary (resp. secondary).  Moving the position of ∞ via motions in S 2 = R 2 ∪ {∞}, we get the table of closed planar curves with # γ ≤ 2 as in Figure 3. For example 1 1 and 1 b 1 (resp. 2 2 , 2 b 2 and 2 c 2 ) are equivalent to [1 1 ] (resp. [2 2 ]) as spherical curves. Here, only the curves of type 1 1 and 2 2 can be drawn with no inflections. Similarly, using the table of spherical curves with # γ ≤ 5 given in [7], we prove the following theorem. The authors do not know of any reference for such a classification of generic locally convex curves.  In particular, the number of equivalence classes of closed locally convex curves with # γ ≤ 5 is 76 (see Figure 3 and Figures 14, 16 and 17 in Section 3). For example, in Figure 3 the curves of type 1 0 , 1 1 , 1 2 , 2 2 satisfy I(γ) = 0, and the Figure 14, the curve of type 6 a 3 is of the same topological type as 6 b 3 as a spherical curve, which is obtained from the 6th curve in the table of curves with # γ = 2 in the appendix of [7].
unless the topological type of γ is as in Figure 4.  Figure 15, the curves of type 6 c 3 consideres as spherical curves are of the same topological type as the curves of type 6 a 3 or 6 b 3 in Figure 14.

Definition of the invariant µ(γ)
We fix a generic closed curve γ : be the inverse image of the crossings of γ, which consists of 2m points in S 1 = R/Z. We set S 1 γ := S 1 \ {c 1 , ..., c 2m }. To introduce the invariant µ(γ), we define special subsets on the curves called 'n-gons': a n , b n ∈ {c 1 , ...c 2m } and the image γ(J) is a piecewise smooth simple closed curve in R 2 . The simply connected domain bounded by γ(J) is called the interior domain of the n-gon. An n-gon is called admissible if at most two of the n interior angles of D are less than π.
We denote by G n (γ) the set of all admissible n-gons, and set Each element of G(γ) is called an admissible polygon. A 1-gon is called a shell (cf. Figure 5). A 2-gon is called a leaf (cf. Figure 6) and a 3-gon is called a triangle  (cf. Figure 7). All shells and all leaves are admissible. However, a triangle whose interior angles are all acute is not admissible.
We fix an admissible n-gon J := [a 1 , b 1 ] ∪ · · · ∪ [a n , b n ]. Then γ(J) is a piecewise smooth simple closed curve in R 2 . We give an orientation of γ(J) so that the interior domain of γ(J) is on the left hand side of γ(J). This orientation induces an orientation on The notions of positivity and negativity of shells were used in [12] and [7] differently from here. The following assertion is the key to proving the inequality (1.2): Proof. Let A 1 , A 2 , · · · , A n be the interior angles of the interior domain of J, and set Γ = γ(J), which we regard as an oriented piecewise smooth simple closed curve with counterclockwise orientation. In this proof, R 2 is considered as the Euclidean plane, and we take the arclength parameter s of Γ. Let s = s 1 , ..., s n be the points where dΓ/ds is discontinuous. We denote by κ Γ (s) (s = s 1 , ..., s n ) the curvature of the curve Γ. Then, the Gauss-Bonnet formula yields that Since J is admissible, we may assume that A 1 , ..., A n−2 ≥ π, and so J κ Γ (s)ds > 0. Then there exist an index i (1 ≤ i ≤ n) and c ∈ [a i , b i ] such that κ Γ (c) > 0. We denote by κ γ (s) the Gaussian curvature of γ at Γ(s). Then we have that which proves the assertion.
We now define the invariant µ(γ) mentioned in the introduction: 1} is called an admissible function of γ if it satisfies the following conditions: is a finite set, and (2) for each J ∈ G(γ), there exists t ∈ supp(Φ) such that t ∈ J and ε J (t) = Φ(t). A point t ∈ supp(Φ) is called a positive point (resp. negative point) if Φ(t) > 0 (resp. Φ(t) < 0). We denote by A the set of all admissible functions of γ.
Then we set Proof of the inequality (1.2). Let γ be a generic closed curve in R 2 . We take a curve σ ∈ γ such that i σ = I(γ). Without loss of generality, we may assume that i γ = I(γ). We can take a point c J ∈ [0, 1] for each J ∈ G in order that sgn(κ γ (c J )) = ε J (c J ), and that c J = c K if J = K. We define a function Φ : Then, Φ is an admissible function. Since i γ = I(γ), the curvature function of γ changes sign at most I(γ) times. So we have that µ(Φ) ≤ I γ , in particular, µ(γ) ≤ I γ . Also we have the following assertion: Proof. Since the number of sign changes of a cyclic sequence of real numbers is always even, µ(γ) is also even. Moreover, if γ has a positive (resp. negative) polygon, each admissible function Φ must take a positive (resp. negative) value. So the existence of two distinct polygons of opposite sign implies that µ(γ) ≥ 2. Now, we prove the converse. A closed curve which is not a simple closed curve 1 0 has at least one shell, and a shell is necessarily a positive or a negative polygon. Suppose that γ has no negative polygons. Then we can choose a point c J ∈ J ∩ S 1 γ for each admissible polygon J ∈ G(γ) such that ε J (c J ) > 0. To show the computability of the invariant µ(γ), we fix 4m (m = # γ ) points t i , s i (i = 1, ..., 2m) satisfying 0 = c 1 < t 1 < s 1 < c 2 < · · · < c 2m < t 2m < s 2m (< 1), and show the following lemma: Lemma 2.6. For each admissible function Φ of γ, there exists an admissible function Ψ satisfying the following properties: (1) supp(Ψ) ⊂ {t 1 , s 1 , ..., t 2m , s 2m }, Proof. We fix an interval U = (c i , c i+1 ), where c 2m+1 := c 1 . If Φ takes non-negative (resp. non-positive) values on U , then we set Since U is arbitrary, we get a function Ψ defined on S 1 γ . Since U ∩ J coincides with either U or an empty set for each J ∈ G(γ), Ψ is an admissible function, and one can easily verify µ(Ψ) ≤ µ(Φ).
Remark 2.7. (Computability of the invariant µ(γ)) The function Ψ obtained in Lemma 2.6 is called a reduction of Φ. (Ψ may not be uniquely determined from Φ, since supp(Φ) ∩ U might consist of more than two points.) By definition, there exists an admissible function Φ such that µ(Φ) = µ(γ). By Lemma 2.6, there is a reduction of such a function Φ. Thus the invariant µ(γ) is attained by a reduced function Ψ. Since the number of reduced admissible functions is at most 3 4m , the invariant µ(γ) can be computed in a finite number of steps.
Remark 2.8. (A flexibility of the reduced admissible function) In the above construction of the function Ψ via Φ we may set Then Ψ is also an admissible function. This modification of Ψ can be done for each fixed interval U = (c i , c i+1 ). However, after the operation, it might not hold that µ(Ψ) ≤ µ(Φ).
Example 2.11. (Curves with a small number of intersections.) Here, we demonstrate how to determine µ(γ) with # γ ≤ 2. In the eight classes of curves in Figure 3, four classes have been drawn without inflections. So I(γ) = 0 holds for these four curves. Each of the remaining four curves satisfies µ(γ) > 0, by Proposition 2.5. On the other hand, these four curves in Figure 3 have been drawn with exactly two inflections. Thus we can conclude that they satisfy µ(γ) = 2. Now, let γ be a curve as in Figure 4. Then γ has four disjoint shells, two of which are positive, and the other two are negative. So we can conclude that µ(γ) ≥ 4. Since γ as in Figure 4 has exactly four inflections, we can conclude that I(γ) = µ(γ) = 4. Example 2.12. (Chain-like curves.) We consider a curve γ with # γ = n (n ≥ 1) as in Figure 8, left. This curve has two shells and n − 1 leaves, including a positive shell and a negative leaf, which are disjoint. Thus (I(γ) ≥)µ(γ) ≥ 2 holds. As in Figure 8, right, this curve can be drawn along a spiral with two inflections. So we can conclude that I(γ) = µ(γ) = 2. In this manner, drawing curves along a spiral is often useful to reduce the number of inflection points. Several useful techniques for drawing curves with a restricted number of inflections are mentioned in Halpern [6, Section 4]. Example 2.13. (Curves with a negative n-gon.) We consider a curve γ n as in Figure 9, which has several positive polygons, but only one negative admissible n-gon, marked in gray in Figure 9. So we can conclude that I(γ) = µ(γ) = 2, as in Figure 9. This example shows that an n-gon (n ≥ 4) is needed to find a curve that cannot be locally convex.
Admissibility of a polygon is important for the definition of the invariant µ. In fact, the curve as in Figure 10, left, has negative polygons which are not admissible, and it can be realized without inflections as in Figure 10, right. Example 2.14. (A curve with an effective leaf which is neither positive nor negative.) We consider a curve γ as in Figure 11. This curve has 5 crossings and exactly two positive shells at A and B. It also has a negative shell at C. Thus µ ≥ 2 by Proposition 2.5. We show that µ(γ) = 4 by way of contradiction: There exist two positive points p 1 and p 2 on the two positive shells at A and B, respectively. There is a unique simple closed arc Γ bounded by A and B which passes through D and E. Suppose that µ(γ) = 2. Then there are no negative points on Γ. Now we look at the negative leaf with vertices A and D. In Figure 11, this leaf is marked in gray. Since Γ does not contain a negative point, there must be a negative point m 1 between A and D on this leaf. Since the curve has a symmetry, applying the same argument to the negative leaf at B and E, there is another negative point m 2 between E and C.
Finally, we look at a leaf with vertices D and E, which is not positive nor negative. Since Γ has no negative point, there must have a positive point p 3 on the arc on the right-hand side of the leaf. Since the sequence p 1 , m 1 , p 3 , m 2 , p 2 changes sign four times, this gives a contradiction. Thus µ(γ) ≥ 4. Since the curve can be drawn with exactly four inflections as in Figure 11, we can conclude that I(γ) = µ(γ) = 4. In this example, an admissible polygon which is neither positive nor negative plays a crucial role to estimate the invariant I(γ) by using µ(γ).
(Proof of the second assertion of Theorem 1.2.) The table of spherical curves up to # γ ≤ 5 is given in the appendix of [7]. By moving the position of ∞ in S 2 = R 2 ∪ {∞}, we get the table of planar curves up to # γ ≤ 5 and can compute the invariant µ(γ). So we can list the curves with µ(γ) = 0. By Proposition 2.5, it is sufficient to check for the existence of positive polygons and negative polygons. When # γ ≤ 2, the number of topological types of such curves is 3. If # γ = 3, 4, 5, then the number of topological types of such curves is 6, 16, 50, respectively. After that we can draw the pictures of the curves by hand. If we are able to draw 76 figures of the curves without inflections, the proof is finished, and this was accomplished in Figures 14, 16 and 17.

Double tangents and geotopical tightness
In this section, we would like to give an application.
Definition 3.1. Let γ be a generic planar curve. We set Then we define which gives the minimum number of double tangents of the curves in the equivalence class γ . (As in Figure 12, d(σ) might be different from d(γ) even if σ ∈ γ and i γ = i σ = I(γ).) A curve σ ∈ γ satisfying d(γ) = δ(γ) is called geotopically tight or g-tight. We call the integer δ(γ) the g-tightness number. Figure 12. Curves of 2 c 2 with i γ = 2 but with different d. The following assertion holds: Proposition 3.2. It holds that δ(γ) ≥ # γ + I(γ)/2.
We expect that any locally convex generic closed curves might be g-tight (see Question 4). Relating this, we can prove the following assertion: Let γ : R → R 2 be a periodic parametrization by arclength of a locally convex curve with total length ℓ, that is, γ(t + ℓ) = γ(t) holds for t ∈ R. Without loss of generality, we may assume that the curvature of γ is positive. Let be a crossing of γ. Replacing s 1 by s 1 + ℓ if necessary, we may assume that s 1 < s 2 andγ(s 2 ) is a positively rotated vector ofγ(s 1 ) through an angle α with 0 < α < π.
When the parameter t varies in the interval [s 1 , s 2 ], the tangent vectorγ(t) rotates through an angle 2πn 1 + α, where n 1 is a positive integer. Similarly, for the interval [s 2 , s 1 + ℓ], there exists a positive integer n 2 such that the rotation angle ofγ(t) is equal to 2πn 2 − α. The sum n 1 + n 2 is the total rotation index of γ. Denote by W (x) the difference n 1 − n 2 . We easily recognize that the sum of W (x) for each crossing x of γ is a geotopy invariant. The following theorem can be proved using the equality of d 2 in [8, p7] (we omit the details): Theorem 3.3. The number of double tangents for any locally convex generic closed curve γ depends only on its geotopy type. More precisely, the following identity holds: where the sum runs over all crossings x of γ.
Remark 3.4. The rotation index R γ of γ which is at each crossing equal to n 1 + n 2 , as mentioned above, is less than or equal to # γ + 1 (cf. [W]). Thus the formula (3.3) implies which reproves Halpern's conjecture in [H2].
In Figure 3, there are two remaining curves of type 1 b 2 and 2 2 , whose g-tightness numbers have not been specified by the authors. The curve of type 2 2 given in Figure 3 satisfies d(γ) = 4, and we expect that δ(γ) = 4. On the other hand, the corresponding curve as in the right of Figure 3 satisfies d = 7, but one can realize the curve with d = 5 as in Figure 1. We expect that it might be g-tight. If true, δ(γ) = 5 holds.
A relationship between inflection points and double tangents for simple closed curves in the real projective plane with a suitable convexity is given in [11]. Finally, we leave several open questions on the invariants I(γ) and δ(γ):  Question 3. Does I(γ) ≤ 2# γ hold for any generic closed curve in R 2 ?
Question 4. Is an arbitrary locally convex curve g-tight?
Question 5. Find a criterion for g-tightness. For example, can one determine δ(γ) when γ is of type 2 2 or 1 b 2 ?