A tropical count of real bitangents to plane quartic curves

A smooth tropical quartic curve has seven tropical bitangent classes. Their shapes can vary within the same combinatorial type of curve. We study deformations of these shapes and we show that the conditions determined by Cueto and Markwig for lifting them to real bitangent lines are independent of the deformations. From this we deduce a tropical proof of Pl\"{u}cker and Zeuthen's count of the number of real bitangents to smooth plane quartic curves.


Introduction
The number of bitangent lines to a smooth plane quartic curve is a classical result from the 19th century. Plücker proved in 1834 that such a curve in the complex projective plane has 28 bitangents [19]. Building on an extensive first count by Plücker [20], Zeuthen proved that a smooth quartic curve can have either 4,8,16 or 28 real bitangents depending on the topology of the underlying real curve in the real projective plane [21]. In this paper we provide a count of real bitangents to a tropically smooth plane algebraic quartic curve using computations and techniques from tropical geometry.
Smooth tropical quartic curves can have exactly 7 or infinitely many bitangent tropical lines grouped into 7 equivalence classes modulo continuous translations that preserve bitangency. This was proven by Baker et al. [2] using the theory of divisors on tropical curves. Bitangents to non-smooth tropical quartics were investigated in [13].
Questions about lifting tropical bitangents were first considered by Chan and Jiradilok [5] for K 4 -curves. Len and Markwig [14] showed that under certain genericity conditions, each class lifts to four bitangent lines over the complex Puiseux series, reproducing Plücker's theorem for tropically smooth quartic curves. These four lifts can be realized by more than one representative in the class and with different multiplicities. Cueto and Markwig [6] proved that each class lifts either zero or four times to real bitangent lines, and they remarked that these real lifts are always totally real, meaning that also the tangency points have real coordinates. As a consequence, every number of real bitangents appearing for a tropically smooth real quartic curve must be divisible by 4. However, a tropical proof of why only the numbers 4, 8, 16 and 28 are observed was still open. The main result of this paper bridges this gap by providing a tropical version of Plücker and Zeuthen's count. Theorem 1. Let Γ be a generic tropicalization of a smooth quartic plane curve defined over a real closed complete non-Archimedean valued field. Either 1, 2, 4 or 7 of its bitangent classes admit a lift to real bitangents near the tropical limit. Every smooth quartic curve whose tropicalization is generic has either 4, 8, 16 or 28 totally real bitangents.
To prove this theorem, we relied on the classification of the combinatorial structure of the bitangent classes by Cueto and Markwig [6]. Smooth tropical quartics are dual to unimodular triangulations of the fourth dilation of the standard 2-simplex. The dual subdivision is also called the combinatorial type of a quartic. The shapes of bitangent classes of tropical quartic curves with the same combinatorial type do not need to be equal, as illustrated in Example 3. This motivates the introduction of deformation classes, which collect for each bitangent class the varying shapes that appear within the same combinatorial type. We provide a classification of deformation classes of tropical bitangents that uniquely depend on the combinatorics of the tropical curve.

Theorem 2.
There are 24 deformation classes of tropical bitangent classes to generic smooth tropical quartic curves modulo S 3 -symmetry. Orbit representatives of their dual deformation motifs are summarized in Figure 11.
The real lifting conditions, i.e., the conditions for admitting a lift to a bitangent over a real closed field with a non-Archimedean valuation, determined by Cueto and Markwig [6], provide us with local information on the number of lifts of each bitangent shape.
The cones in the secondary fan of 4∆ 2 that induce unimodular triangulations parameterize tropically smooth plane quartics. Bitangent shapes can vary within a given cone, since they depend on the edge lengths of the tropical quartic curve, rather then solely on the input triangulation. However, Theorem 14 states that the lifting conditions over a real closed complete non-Archimedean valued field depend solely on the triangulations. This motivates the construction of deformation classes of bitangent shapes, which are determined by the given triangulation and provide explicit sign rules for lifting each deformation class (see Table 5).
Using polymake [8], we enumerate the deformation classes for every combinatorial type of tropical quartic curves. We then check the real lifting conditions obtaining the expected number of real bitangent lines. We impose the same genericity assumptions as in [6,14] and we discuss them further in Section 2. In particular, they include the smoothness of the tropicalized curve. Note that the count over the reals follows from working over a real closed field by Tarski-Seidenberg's Transfer Principle [3,Theorem 1.4.2]. Real tropicalization goes back to work of Maslov [17] and the study of logarithmic limit sets of (real) algebraic varieties, see also [1]. This paper is organized as follows. In Section 2, we introduce the main definitions and we report the classification of tropical bitangent classes and their lifting conditions, as introduced in [6]. We assume that the reader is familiar with basic definitions and results on tropical curves and regular triangulations. We refer to [7,15] for further details. Deformation classes are defined and classified in Section 3. In Section 4, we study their real lifts and prove Theorem 1. The proof is based on the enumeration of deformation classes in polymake. The technical description of the algorithms and their implementation can be found in [10]. The proofs for the classification are always constructed by providing details of few cases, and then summarizing the main ideas of the remaining ones. More examples and figures are collected in the Appendix A in order to provide further geometric intuition for the definitions and proofs.

Tropical quartic curves and their bitangents
A plane quartic curve V (f ) is the zero set of a polynomial of degree four f (x,y) = a 00 + a 10 x + a 01 y + a 20 x 2 + a 11 xy + a 02 y 2 + a 30 x 3 + a 21 x 2 y (1) + a 12 xy 2 + a 03 y 3 + a 40 x 4 + a 31 x 3 y + a 22 x 2 y 2 + a 13 xy 3 + a 04 y 4 .
We consider the tropicalization of curves defined over a real closed complete non-archimedean valued field K R and its algebraic closure K. For the tropicalization of curves, we use the max-convention. We write λ ij for the valuations of the coefficients of the polynomial, i.e., λ ij = val(a ij ). Then, Trop(V (f )) is the tropical curve define by the tropical polynomial with coefficients −λ ij .
We assume that the Newton polygon of f is the fourth dilation of the standard 2dimensional simplex 4∆ 2 . The polygon 4∆ 2 contains 15 lattice points p ij corresponding to the monomials x i y j of f . By duality, the combinatorics of the tropical curve Γ = Trop(V (f )) is determined by the subdivision T (of the lattice points) of 4∆ 2 induced by the coefficients −λ ij . We use the notation · ∨ to refer to the dual of a vertex or an edge of Γ in T , and viceversa. We only consider smooth tropical plane quartic curves, so the subdivisions of 4∆ 2 are unimodular triangulations. In particular, all lattice points in 4∆ 2 are vertices in the triangulation T . The set of points in R 15 inducing the same subdivision is a relative open cone called the secondary cone and denoted Σ(T ). For a point c ∈ Σ(T ), we use the notation Γ c to indicate the tropical quartic curve defined by the tropical polynomial with coefficients given by the coordinates of c.
Regular unimodular triangulations of 4∆ 2 have been enumerated by Brodsky et al. in [4]. They counted 1278 orbits of combinatorial types under the action of the symmetric group S 3 . This group acts on the homogenization of the lattice points of 4∆ 2 and on the corresponding monomials of the polynomial f and its tropicalization Trop(f ).
A tropical line Λ is bitangent to a smooth tropical plane quartic curve Γ if their intersection Λ ∩ Γ has two components with stable multiplicity 2, or one component with stable multiplicity 4. See [15,Section 3.6] for an introduction to stable intersections. We always assume that tropical bitangent lines are non-degenerate, i.e., each tropical line consists of a vertex and three adjacent rays with directions −e 1 , −e 2 and e 1 + e 2 given by the standard basis of R 2 . For an impression of tropical bitangent lines, see Figure 3. A tropical quartic curve Γ has exactly 7 or infinitely many tropical bitangents. The collection of bitangents can be grouped into 7 equivalence classes modulo continuous translations that preserve bitangency as shown in [2]. Formally, the tropical bitangent classes of a tropical quartic Γ are defined as the connected components of R 2 containing the vertices of tropical bitangents in the same equivalence class. Up to S 3 -symmetries, they refine into 41 shapes of tropical bitangent classes given by coloring the points in the class belonging to Γ, see Figure 2. Since a non-degenerate tropical line is determined by its vertex, bitangent classes formally live in the dual plane. They are connected polyhedral complexes, which are also min-tropical convex sets [6, Theorem 1.1, Corollary 3.3]. To improve the visualization, we draw the curve and its bitangent classes on the same plane.
As remarked by Cueto and Markwig [6], the shapes of tropical bitangent classes of a smooth tropical quartic curve Γ impose combinatorial constraints on the regular unimodular triangulation of 4 ∆ 2 dual to Γ. More precisely, the existence of a representative of a certain shape determines specific subcomplexes that must be contained in the triangulation, see Figure 4. Such a subcomplex is only a necessary condition for the presence of its corresponding tropical bitangent shape. Given a smooth tropical quartic curve Γ, the shapes of its bitangent classes are not fully determined by the combinatorial type of Γ, but they also depend on the length of the edges, as the following example illustrates. This motivates us to introduce deformation classes of tropical bitangents in the following section.
Example 3. We consider the quartic curve dual to the triangulation T in Figure 3a. The colored subcomplex of T corresponds to shapes (E), (F) and (J) in the classification in Figure 2. Let a denote the coefficient vector of an algebraic curve of degree 4 with entries ordered as in (1).
The number of such 's is the lifting multiplicity of Λ. In an equivalence class, the number of tropical bitangents which lift and their lifting multiplicities can be one, two or four, see [14,Theorem 4.1].
Similarly, we are interested in the number of real lifts, that is, the lifting multiplicity when V (f ) and are defined over K R . In this case, as shown by Cueto and Markwig [6], every bitangent class of a given shape has either zero or four lifts to real bitangents. Moreover, the existence of a lift uniquely depends on the signs s ij of the coefficients a ij of the polynomial f . Table 3 in Section 4 summarizes these conditions.
Lifting problems were studied under the genericity constraints explained in [6, Remark 2.10], which also apply in our results next to the assumptions that the tropical curve Γ is smooth and that the tropical bitangent lines are non-degenerate. The assumption that if the tropical curve Γ contains a vertex adjacent to three bounded edges with directions −e 1 , −e 2 and e 1 + e 2 , the shortest of these edges is unique will be particularly relevant in our analysis of the lifting conditions of shape (C).
We conclude this section by fixing some notation and conventions. We follow the labeling of shapes of bitangent classes and the color patterns introduced in [6] by Cueto and Marking, see Figures 2 and 4. The group S 3 acts on bitangent shapes and their dual subcomplexes. We refer to the bitangent shapes and subcomplexes in Figures 2 and 4 as in identity position. We indicate the different elements in the orbit of a bitangent shape not in identity position by adding the element of S 3 acting on it to the index of the shape. For example, the notation (B) (xy) means that the bitangent class has shape (B) with dual complex given by the action of (x y) on the complex in identity position.

Deformation classes of tropical bitangents
In this section we introduce deformation classes of tropical bitangents and we classify them. Our terminology is inspired by the one introduced in [11,18] for the classification of the combinatorial positions of tropical lines on cubic surfaces.

Definition 4.
Let Γ be a tropical smooth quartic curve with dual triangulation T , and let B be a bitangent class of Γ of a fixed shape. The dual bitangent motif of (Γ, B) is the subcomplex of T that is fully and uniquely determined by the shape of B and its position with respect to Γ. Dual bitangent motifs are classified in [6, Figure 19].
In pictures of dual bitangent motifs, we use the same color coding as Cueto and Markwig [6]. Black and colored solid edges must be part of the triangulations, while dotted ones represent possible edges, of which one must occur. Black vertices are always present,  Figure 19].
while the colored ones are endpoints of colored possible edges or they form a triangle with an edge of the same color. Different colors correspond to different types of tangencies, see [6,Remark 4.13].
The summary of the dual bitangent motifs as shown in Figure 4 is very condensed, and, as a consequence, for some figures with dotted edges not all combinations lead to the assigned bitangent shape. For example, when choosing the red and green edges p 10 p 11 and p 01 p 11 in the picture of (A), the shape, which will occur, will be either (S) or (P), but never (A). Therefore, the dual bitangent motifs will be subdivided more in the following classification of the deformation classes.

Definition 5.
Given a tropical quartic Γ c with dual triangulation T , c ∈ Σ(T ), and a tropical bitangent class B, we say that a tropical bitangent class B is in the same deformation class as B if the following conditions are satisfied: There exists Γ c with c ∈ Σ(T ) having B as one of its bitangent classes.
There is a continuous deformation from Γ c to Γ c given by a path in the secondary cone Σ(T ) from c to c that induces B to change to B .
We use the notation B ω to indicate the deformation of B in Γ ω for ω in the path. Given a unimodular triangulation T of 4∆ 2 and a dual quartic curve Γ, let D be the deformation class of one of its seven bitangent classes. The dual deformation motif of (T ,D) is the union of the dual bitangent motifs of all shapes belonging to bitangent classes in D.
We label deformation classes using the letters of the shapes of tropical bitangents. In Example 3 we saw a deformation class (E J F). If the class contains the image of shapes under the action of an element σ ∈ S 3 , we use the notation +σ. We remark that σ acts on all shapes in the deformation class, but some of them might be invariant. For example, the deformation class (B H' H)+(yz) consists of the shapes (B), (H'), (H) = (H) (yz) , (H') (yz) and (B) (yz) . Remark 6. Each smooth tropical quartic Γ has 7 deformation classes, and it follows from the definition that they only depend on the dual triangulation T of Γ. Changing the coefficients defining Γ in the secondary cone Σ(T ) induces a variation in the shapes of the tropical bitangents within the deformation class.
In order to classify deformation classes, we need to distinguish between the different subcomplexes which give a bitangent shape, e.g. the two different choices of blue edges for shape (E). We have to do this because the combinatorial type of the curve determines how the shapes can deform. Proof. The proof works similarly for each of the cases. The main argument for each case is that due to the combinatorial structure of T , the two tangencies cannot change the type of their intersection, and this fully determines the bitangent class and its shape. We explain the details for shape (E) and summarize the remaining cases in Table 1. Let B be a bitangent class of shape (E). We may assume that the dual bitangent motif looks as in Figure 6. One tangency point is a transversal intersection in the edge of Γ dual to E = p 20 p 13 . The second tangency point is a non-transversal intersection of the horizontal ray of Λ with the bounded edge of Γ dual to p 10 p 11 or p 11 p 12 . We can exclude the edge p 12 p 13 since it does not define a dual bitangent motif of shape (E).
The bitangent class B is a line segment. Its two endpoints are determined by intersecting rays with direction −(e 1 + e 2 ) through the vertices of the edge E ∨ of Γ with a ray with direction e 1 passing through the non-transversal intersection. See Figure 6 for an example. Independently of the lengths of the edges of Γ, the endpoints of B cannot lie in Γ because of the slopes of the edges of Γ that connect the vertices ∆(p 10 p 11 p 20 ) ∨ and ∆(p 12 p 13 p 20 ) ∨ . Table 1 contains the remaining bitangent classes from the statement. For each deformation class, we draw the dual deformation motif in the dual triangulation and the relevant part of the tropical quartic curve. When there are several possible dual bitangent motifs condensed in a picture, we draw one tropical curve dual to only one of them. From the combinatorial shape of the tropical curve, we can see that changes of edge lengths cannot induce a deformation of the shape of the bitangent class.   shape. We now consider two cases of deformation classes of not constant shape. Further details on the proofs of the upcoming lemmas can be found in [9].  Proof. Let B be a bitangent class with dual bitangent motif as in Figure 7. The first tangency point is the transversal intersection of the diagonal ray of the bitangent with the edge of Γ dual to E = p 20 p 31 . The second tangency point is a non-transversal intersection with one of the three bounded edges of Γ dual to p 10 p 11 , p 11 p 12 or p 12 p 13 . Since T is unimodular, we remark that the subcomplexes we are considering contain the dual bitangent motifs of shapes (E), (F) and (J) in identity position. We need to show that for any Γ c with c ∈ Σ(T ) the bitangent class B can only deform between these shapes.
Let E be the vertical red dotted edge in T that forms a triangle with the red circled lattice point p 20 . The bounded edge of Γ dual to E will always have y-coordinate smaller than the vertex v = ∆(p 20 p 31 p 21 ) ∨ . Hence, for any Γ c the tangent points of the bitangent class are contained in the edge dual to E and the bounded edge dual to E .
Changes in the length of the edge dual to E influence the position of the intersections. Depending on the y-coordinate of the bounded edge dual to E in comparison with the vertex v, we obtain shape (E), (J) or (F) for B. This is illustrated in Figure 8 for the case E = p 12 p 13 . The other two cases are analogous. We cannot obtain another shape for B because the value of the x-coordinate of the points in the edge dual to E and of v do not influence the shape.  Proof. We distinguish the two cases in Figure 9 depending on whether the edge p 11 p 22 or the edge p 11 p 04 is contained in the triangulation T . Figure 10 shows deformations of B into the claimed shapes by edge length changes of Γ when the blue edge is E = p 11 p 22 . Analogous pictures can be drawn for p 11 p 04 . The edge lengths can be interpreted in terms of relative positions of vertices of Γ. This approach is considered in the proof of the classification of bitangent shapes in [6,Section 4]. It remains to argue that these are the only shapes B can deform into. We see that B cannot deform into any other shapes by considering Figure 10. Rays with direction e 1 + e 2 passing through the vertices of E ∨ cannot intersect the upper vertex of (p 01 p 11 ) ∨ or the right vertex of (p 10 p 11 ) ∨ . Thus, B cannot deform further.
Similar reasoning leads us to complete the classification in Theorem 2. Orbit representatives of the dual complexes describing the deformation classes are illustrated in Figure 11.
Proof of Theorem 2. The general idea behind the proof is as follows: We look at the classification of dual bitangent motifs summarized in Figure 4. We identify common subcomplexes in the dual bitangent motifs and check whether deformations between corresponding bitangent shapes are possible. Lemma 7 describes deformation classes with constant shapes, and Lemmas 8 and 9 cover deformation classes (E J F) and (G K U U' T T' T" V)+(x y). The remaining cases are summarized in the following table. Figures of the deformation classes are provided in Appendix A. For details see [9]. No further deformations: The lower vertex of (p 21 p 30 ) ∨ has always smaller y-coordinate than ∆(p 1v p 1,v+1 p 21 ) ∨ , and the upper vertex of (p 21 p 31 ) ∨ has always larger y-coordinate than (p 1v p 1,v+1 ) ∨ .
(D L' Q) First tangency: Transversal intersection of the diagonal, vertical ray of the bitangent line with (p 11 p 22 ) ∨ , (p 22 p 10 ) ∨ , respectively. Second tangency: Depending on the edge lengths, the non-transversal intersection with (p 10 p 11 ) ∨ in the horizontal ray of the bitangent line (D) deforms to a transversal intersection of (p 00 p 11 ) ∨ with the diagonal ray (Q), through the shared vertex (L'). No further deformation: Due to the edge directions determined by the dual bitangent motif.

(D L' Q Q' R)
First tangency: Same as for (D L' Q). Second tangency: Depending on the edge lengths, the non-transversal intersection with (p 10 p 11 ) ∨ in the horizontal ray of the bitangent line (D) deforms through the shared vertex (L') to a transversal intersection of (p 00 p 11 ) ∨ with the diagonal ray (Q), through the shared vertex (Q') to a non-transversal intersection of (p 01 p 11 ) ∨ with the vertical ray (R). No further deformation: The upper vertex of (p 01 p 11 ) ∨ has always larger y-coordinate than any intersection of the ray of direction −(e 1 + e 2 ) starting at ∆(p 22 p 11 p 12 ) ∨ with (p 01 p 11 ) ∨ .
(D L O) First tangency: Same as for (D L' Q). Second tangency: Depending on the edge lengths, the non-transversal intersection with (p 10 p 11 ) ∨ in the horizontal ray of the bitangent line (D) deforms through the shared vertex (L) to a non-transversal intersection of (p 01 p 11 ) ∨ with vertical ray (O).

No further deformation:
Same as for (D L' Q Q' R).

(G I N)+(x y) = (G I N I (xy) G (xy) )
First tangency: Transversal intersection of the diagonal ray with (p 11

Real lifting conditions of deformation classes
In this section, we focus on the real lifting conditions determined by each shape in a deformation class. This leads to a new proof of Plücker and Zeuthen's count for tropically smooth quartics. First, we focus on the two deformation classes discussed in Lemmas 8 and 9. Then, we analyze the sign rules for the deformation class (C) which requires a special argument, as is explained in Example 12. There are six cases to treat depending on the relative order of the edge lenghts of three relevant bounded edges of Γ yielding shape (C). Sign rules for lifting the remaining deformation classes are provided in Theorem 14. Table 5 contains a summary of our fundings.
Our starting point are the real lifting conditions determined by Cueto and Markwig [6] and reported in Table 3. With the notation s mn we indicate the sign of the coefficient c mn of the monomial corresponding to the lattice point p mn . In the proofs, we will consider five parameters i, j, k, u and v. The edge p u1 p u+1,1 is the green edge in a dual bitangent motif. The parameter j specifies the point p j0 forming a triangle with the green edge. The edge p 1v p 1,v+1 is the red edge in a dual bitangent motif. The parameter i specifies the point p 0i forming a triangle with the red edge. Finally, k parametrizes a blue edge p 11 p k,4−k .

Shape
Lifting   Proof. From the proof of Lemma 8 we know that a bitangent class in the deformation class (E J F) can deform into any of the three shapes, all in the same position with respect to the action of S 3 . As the triangulation T is fixed, the values that have to be substituted for v and i in the formula from Table 3 do not change for the three shapes.

Lemma 11. Let Γ be a smooth tropical plane quartic curve with dual triangulation T and a bitangent shape B in a deformation class (G K U U' T T' T" V)+(x y).
For every c ∈ Σ(T ) the real lifting conditions of B c in Γ c are independent of the shape of the bitangent class.
Proof. Figure 9 shows the two dual deformation motifs of the deformation class. The lifting conditions of shape (G) and shapes (K), (U), (U'), (T), (T'), (T"), (V) in identity position differ by the factor (−s 10 s 11 ) i . Since the vertex relevant to the value of i is p 00 , the value of i is 0, so the two conditions coincide. This value does not change under the (x y) permutation. The sign s k,4−k is uniquely determined by the dual motif, so again it stays the same for all shapes. Thus, we can conclude for both dual deformation motifs in Figure 9 that the real lifting conditions are independent of the shapes.
We now focus on the special case of deformation class (C). In [6], the lifting conditions for the bitangent class (C) are computed for generic tropical quartics satisfying the following condition: If Γ contains a vertex v adjacent to three bounded edges with directions −e 1 , −e 2 and e 1 + e 2 , then there exists a unique shortest edge. The vertex of a tropical bitangent of shape (C) coincides with v. Cueto and Marking chose the edge with direction −e 2 as shortest edge. We denote with λ 1 , λ 2 and λ 3 the lattice lengths of the edges adjacent to the vertex v with direction −e 2 , −e 1 and e 1 + e 2 , respectively. When λ 1 < λ 2 λ 3 , we call this the identity case of the genericity condition.
Any generic tropical quartic having a bitangent class of shape (C) at a vertex v, but with different edge lengths, can be brought into this position by applying an action of S 3 . This changes the dual subdivision accordingly and, as consequence, also the formula for the real lifting conditions of (C). We illustrate this in Example 12. For some tropical quartic curves with bitangent class (C), the genericity condition of a unique shortest edge adjacent to v can never be satisfied. This happens when the dual triangulation only allows a pair of shortest edges adjacent to v. Up to S 3 -action, these triangulations are unimodular refinements of the partial subdivision in Figure 12.
Example 12. We consider the two smooth tropical quartic curves dual to the triangulation T shown in Figure 13. These quartic curves have a bitangent class of shape (C). For the tropical curve in Figure 13b, the lengths λ 1 , λ 2 , λ 3 satisfy the identity case of the genericity condition: λ 1 < λ 2 λ 3 . We substitute i = 2, j = 1, k = 2 in the real lifting condition for (C) in Table 3   By choosing a different weight vector in Σ(T ), we can deform the edge lengths such that λ 2 < λ 1 λ 3 . An example of this is shown in Figure 13c. In this case, we are no longer in the identity case, so in order to apply the lifting formula, we need to apply the action of (x y) to switch the lengths λ 1 and λ 2 , inducing also an action on the triangulation T . The image of the curve and of T under (x y) is depicted in Figure 14. Now, we have to substitute i = 1, j = 2, k = 2 in the lifting conditions for shape (C) obtaining    Figure 13c and its dual triangulation. We have i = 1, j = 2, and k = 2.
We then deduce the lifting conditions for the original quartic with λ 2 < λ 1 λ 3 by applying (x y) −1 = (x y): − s 11 s 21 s 10 s 02 > 0 and s 22 s 02 > 0. ( The second inequalities in (2)  Proof. We fix the following notation, see also Figure 15: i is the y-coordinate of the vertex p 0i , which forms a triangle with the (red) edge p 11 p 12 , j is the x-coordinate of the vertex p j0 , which forms a triangle with the (green) edge p 11 p 21 and k is the x-coordinate of the vertex p k,4−k , which forms a triangle with the (pink) edge p 12 p 21 . We compute the real lifting conditions of shape (C) for a bitangent class not in identity position. Suppose that Γ c has a unique shortest edge among λ 1 , λ 2 , λ 3 . Then, there exists σ ∈ S 3 such that for σ(Γ) the lattice lengths of the edges adjacent to σ(B) satisfy λ 1 < λ 2 λ 3 . This corresponds to σ(B) being in identity position. We can then determine the real lifting condition for σ(B) using Table 3 and the parameters from σ(T ). In order to do this, we first need to look at the images of the three lattice points p 0i , p j0 and p k,4−k under σ. Their images will lie in the boundary of 4∆ 2 : σ(p 0i ), σ(p j0 ), σ(p k,4−k ) ∈ {p 0ĩ , pj 0 , pk ,4−k }. Secondly, we substitute the values of the tilde indices into the lifting conditions, and then apply σ −1 to obtain the real lifting conditions of B = σ −1 (σ(B)) in Γ = σ −1 (σ(Γ)). Finally, we have to compare the lifting conditions of the bitangent class B of shape (C) in Γ with the ones of B c of shape (C) in Γ c where c ∈ Σ(T ) such that the dual deformation motif of (Γ c ,B c ) is in identity position and Γ c satisfies λ 1 < λ 2 λ 3 .
The sign rules in the table are obtained by applying the corresponding permutation to each formula for the identity position (row 1 of the table) and replacing the values of i, j, k by the corresponding ones. The procedure should be clear from Figure 1. For example, when applying (xyz), the values of (i, j, k) should be replaced with (j, k, i).
Then, it suffices to use the same techniques as in Example 12 to confirm the sign rules for all 6 cases are equivalent. For example, to show the equivalence between cases 1 and 2, it is enough to check that the first inequalities in both cases agree, whereas replacing s j0 by (−s 11 ) i+j s i 12 s j 21 s 0i (which follows from the first inequality) in the second inequality for case 1 yields the remaining inequality in case 2. The equivalence between case 1 and the remaining cases follows by similar reasoning since they always share a common inequality.  Table 3.

(B H' H), (B H' H)+(y z)
The dual bitangent motif of (B) yields j = 4. Substituting it gives the same real lifting conditions for (B), (H') and (H). The lifting conditions for (H) and (H) (y z) are the same, so the statement follows by symmetry.

(B M)+(y z)
The dual bitangent motif of (B) yields j = 3. Substituting it gives the same real lifting conditions for (B) and (M). The real lifting conditions for (M) and (M) (y z) are the same.

(D L' Q), (D L' Q Q' R)
The dual bitangent motif of shape (D) yields i = 0, so the real lifting conditions for all shapes in the two deformation classes coincide.

(D L O)
The dual bitangent motif of shape (D) yields i = 1, so the real lifting conditions for all shapes in this deformation class coincide.

(G I N)+(x y)
The dual bitangent motif of shape (G) yields i = 1, so the real lifting conditions of (G), (I) and (N) coincide. Also, (N) and (N) (x y) have the same real lifting conditions.
Deformation class parameters

(G K U T T')
The dual bitangent motif of shape (G) yields i = 0, so the real lifting conditions for all shapes in this deformation class coincide.
Remark 15. Markwig, Payne and Shaw characterize the lifting conditions of bitangent shapes over more general fields [16]. An analogous investigation to the proof of Theorem 14 shows that the lifting conditions again stay constant in every deformation class. Therefore, deformation classes are relevant for the lifting behavior of tropical bitangents over other fields, not only over real closed fields. Table 5 summarizes the real lifting conditions for the deformation classes described in Lemmas 10, 11, Proposition 13 and Theorem 14.  Table 5: Real lifting conditions of the deformation classes in their positions as in Figure 11.
We are now ready to give a proof of our main result. Proof of Theorem 1. Let Γ be a generic tropicalization of a tropically smooth quartic curve V (f ) defined over K R and T its dual triangulation. By Theorem 14, the real lifting conditions of the 7 bitangent classes of Γ only depend on their 7 deformation classes. Furthermore, the deformation classes are uniquely determined by their dual deformation motifs in the triangulation T , as pointed out in Remark 6. Therefore, the real lifting conditions for the tropical bitangent classes of Γ only depend on the triangulation T .
In order to prove the statement, we need to enumerate the dual deformation motifs of the deformation classes in the 1278 unimodular regular triangulations of 4∆ 2 modulo S 3 -symmetry as computed in [4]. Of these S 3 -representatives exactly eight do not satisfy the genericity constraint that a vertex of the curve with adjacent edges of directions −e 1 , −e 2 , e 1 + e 2 needs to have a unique shortest adjacent edge, see Figure 12. For these eight cones we did run the same computations as for the generic cones, but we could not compute the lifting behavior of the bitangent class of shape (C) since this is not yet understood. However, since the numbers of real bitangents are already known classically, our computations might help to understand these special cases.
We implemented the search for the dual deformation motifs of the deformation classes in polymake [8]. For each deformation class, we considered the real lifting conditions determined in Theorem 14 and summarized in Table 5, and we evaluated them for all possible 2 15 sign vectors. Again, we implemented this in polymake obtaining that each sign vector satisfies the lifting conditions of 1, 2, 4 or 7 deformation classes. A more detailed description of the computational procedure and codes can be found in [10].
Finally, by [  Of the 15 coefficients of the lifted algebraic curve, the maximal number of 12 is involved in the sign conditions. We fix all signs s ij positive except for s 10 , s 21 and s 03 . When varying these three signs, different lifting conditions will be satisfied, and consequently, the lifted real algebraic quartic curves will have different numbers of bitangent lines as summarized in the table below. The topological type of the curves near the tropical limit is also determined by the signs and can visualized via patchworking in polymake [12].

A.1 Example of deformation class (B H' H)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 16. Note that we are choosing the red edge p 12 p 13 in the corresponding picture in Figure 11. A similar example can be drawn for p 11 p 12 .

A.2 Example of deformation class (B H' H)+(y z)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 17. Note that we are choosing the red edge p 12 p 13 in the corresponding picture in Figure 11. Similar examples can be drawn for p 10 p 11 and p 11 p 12 .

A.3 Example of deformation class (B M)+(y z)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 18. Note that we are choosing the red edge p 12 p 13 in the corresponding picture in Figure 11. Similar examples can be drawn for p 10 p 11 and p 11 p 12 .

A.4 Example of deformation class (D L' Q)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 19.

A.5 Example of deformation class (D L' Q Q' R)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 20.

A.6 Example of deformation class (D L O)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 21. The situation is similar if we choose the other blue edge p 11 p 04 , as illustrated in Figure 23.

A.8 Deformation class (G K U T T')
As for deformation class (G I N)+(x y) there are two different cases of how the second tangency arises. Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 24. This figure depicts the case where one tangency is given by the blue edge p 11 p 22 .

A.9 Deformation class (W X (xz) Y (xz) EE GG)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 26. Figure 26: Example of deformation class (W X (xz) Y (xz) EE GG).

A.10 Deformation class (W. . . HH)+(x z)
Let Γ be a smooth tropical quartic curve with a bitangent class B with dual bitangent motif in identity position contained in the subcomplex in Figure 27.