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The contact mapping class group and rational unknots in lens spaces

We determine the contact mapping class group of the standard contact structures on lens spaces. To prove the main result, we use the one-parametric convex surface theory to classify Legendrian and transverse rational unknots in any tight contact structure on lens spaces up to Legendrian and transverse isotopy.


INTRODUCTION
Ever since Eliashberg [13] determined the homotopy type of the group of contactomorphisms of ( 3 , ) relative to a point, there have been several studies on the group of contactomorphisms of various contact manifolds. For example, Gompf [31] showed that there exists a contactomorphism of the standard contact structure on 1 × 2 which is smoothly isotopic to the identity, but not contact isotopic to the identity. After that, there have been many similar results, see [5,10,22,25,39] for examples. Also, there have been studies on higher homotopy groups, see [4,5,6,14,19,20] for examples.
However, the contact mapping class group has not been determined for many contact manifolds so far. If we focus on closed manifolds, the contact mapping class group was only determined for the standard contact structure on 3 by Eliashberg [13], the overtwisted contact structures on 3 by Vogel [42], and the canonical contact structures on the unit cotangent bundles * Σ for ≥ 1 and their cyclic covers by Giroux and Massot [29]. The main difficulty to study the contact mapping class group is that we need to understand the behavior of some Legendrian (or contact) submanifolds under Legendrian (or contact) isotopy, which has been poorly studied in other than 3 . According to the author's knowledge, the only known results on the classification of Legendrian knots outside of 3 were the linear curves in any tight contact structure on 3 by Ghiggini [24] and some knots and links in the standard contact structure on 1 × 2 by Ding and Geiges [11] and by Chen, Ding and Li [8].
In Theorem 1.1 and 1.3, we determine the contact mapping class group of the standard contact structures on lens spaces and 1 × 2 . The main ingredients are Theorem 1.5 and 1.6, the classification of Legendrian and transverse rational unknots up to Legendrian and transverse isotopy. Once we have the classification, we can perturb a contactomorphism to fix a standard neighborhood of some Legendrian rational unknot. Then the problem reduces to determine the contact mapping class group of its complement, which 1 is determined in Theorem 3.1 by applying the one-parametric convex surface theory, in particular, Colin's isotopy discretization [9], and various properties of bypasses studied by Honda [34,35] and Honda, Kazez and Matić [37], see Section 2. 2 The main technique for Theorem 1.5 and 1.6 is again the application of one-parametric convex surface theory and various properties of bypasses, which were utilized to study Legendrian and transverse knots in contact structures on 3 in [7,18]. In Section 3, we develop the technique and apply it to a solid torus and lens spaces to study Legendrian and transverse knots in tight contact structures on those manifolds.
1.1. The contact mapping class group of lens spaces. We first review the basic notations. First, we assume every contactomorphism is a coorientation preserving one unless otherwise specified. We denote the group of contactomorphisms of a closed contact manifold ( , ) by Cont( , ) = the group of coorientation preserving contactomorphisms of ( , ).
The contact mapping class group of ( , ) is defined to be the group of contact isotopy classes of contactomorphisms of ( , ). We denote it by 0 (Cont( , )) = Cont( , )/∼ where ∼ if is contact isotopic to .
Let be the unknot in 3 . For a pair of coprime integers ( , ) satisfying > > 0, we define ( , ) = 3 − / ( ). Now we are ready to state our first main result. For the first two cases, the contact mapping class group is generated by a contactomorphism . See Section 2.3 for the definition of .
There is a quick application of Theorem 1.1. In the forthcoming paper with Baker, Etnyre, and Onaran [2], we classify Legendrian and transverse torus knots in the standard tight contact structures on lens spaces. Basically, we classify the knots coarsely, meaning that up to coorientation preserving contactomorphism which is smoothly isotopic to the identity. However, due to Theorem 1.1, we could improve the result up to Legendrian and transverse isotopy.
Comparing the contact mapping class group and the smooth mapping class group of lens spaces (Theorem 2.11), we can make the following observation.

Legendrian and transverse rational unknots in lens spaces.
A rational unknot in a lens space is a core of a Heegaard torus. See Section 2.4 for more details, and also see Figure 1 and 7 for (contact) surgery presentations of rational unknots in lens spaces.
Etnyre and Baker [1] coarsely classified Legendrian rational unknots in any tight contact structure on lens spaces, see Theorem 2.18. Recall that the coarse classification is the classification up to contactomorphism which is smoothly isotopic to the identity. Also, Geiges and Onaran [23] coarsely classified non-loose Legendrian unknots in some lens spaces.
However, to study the contact mapping class group, we need to understand the behavior of Legendrian (or contact) submanifolds under Legendrian (or contact) isotopy. Our second main result is the classification of Legendrian rational unknots in any tight contact structure on lens spaces up to Legendrian isotopy. where / is the largest rational number satisfying − = −1. Also the rational rotation numbers are determined by the formula in Lemma 2.19 or 2.20. Every Legendrian representative of rational unknots in is Legendrian isotopic to one of the Legendrian representatives above, or their stabilization.
See Figure 2 for the Legendrian mountain range of the rational unknot in (2, 1). We also give the classification of (positive) transverse rational unknots in any tight contact structure on lens spaces up to transverse isotopy. Theorem 1.6. Suppose > > 0 and is a tight contact structure on ( , ). Rational unknots in are transversely simple: there are transverse representatives such that every transverse representative of rational unknots in is transversely isotopic to one of the transverse representatives above, or their stabilization. Also. is a positive transverse push-off of and is a positive transverse push-off of − . There is an application of Theorem 1.5 and 1.6. In the forthcoming paper with Baker, Etnyre, and Onaran [2], we classify Legendrian and transverse positive torus knots in any tight contact structure on lens spaces. Basically, we classify the knots coarsely. However, due to Theorem 1.5 and 1.6, we could improve the result up to Legendrian and transverse isotopy. Unfortunately, there are some subtleties for negative torus knots (e.g. Legendrian large cables, see [7]), so we could only improve the result for the negative torus knots with sufficiently negative tb ℚ and sl ℚ .
Acknowledgements. The author thanks Anthony Conway for helpful comments on the first draft of the paper and John Etnyre for a useful conversation.

BACKGROUND AND PRELIMINARY RESULTS
In this section, we review and prove some useful results on contact topology and the mapping class group of lens spaces that will be used throughout the paper. We assume the reader is familiar with 3-dimensional contact topology, in particular, Legendrian and transverse knots and the convex surface theory. See [16,21,33] for more details.

Convex surfaces and bypasses.
First, we warn the reader that our convention is slopes of curves on a torus are given by meridian longitude . This has led to some differences between how we cite results and how they were initially stated.
We will use several properties of convex surfaces without explicitly mentioning them: perturbing a compact surface to be convex, realizing a particular characteristic foliation for the given dividing set, and using the Legendrian realization principle. We also assume that the boundary of any convex surface Σ is Legendrian, if non-empty. When Σ is connected, then Σ is null-homologous and ( Σ) is well-defined. Kanda [38] proved where Γ Σ is the dividing set of Σ. Suppose Σ is a properly embedded convex surface in a contact 3-manifold with convex boundary. Then the relative Euler class of the contact structure evaluates to (Σ + ) − (Σ − ) on Σ where Σ ± are the positive/negative regions of the convex surface. We can modify a convex surface by attaching a bypass, introduced by Honda [33]. Consider a convex overtwisted disk whose dividing set consists of a single contractible closed curve. Take a properly embedded arc on the disk intersecting the dividing curve in two points. By applying the Legendrian realization principle, we can assume that is a Legendrian arc, and cut the disk along ; each half-disk is called a bypass. Now, suppose a bypass transversely intersects a convex surface Σ such that ∩ Σ = . Let Γ Σ be the dividing set of Σ. Since the dividing set interleaves, intersects Γ Σ in three points. We call the Legendrian arc on Σ the attaching arc of the bypass and say is a bypass for Σ. After edge-rounding, the convex boundary of a neighborhood of ∪ Σ is a surface isotopic to Σ but with its dividing set changed in a neighborhood of the attaching arc as shown in Figure 3. We call this process a bypass attachment along . Note that Figure 3 is drawn for the case that the bypass is attached "from the front", that is, sitting above the page. If we attach a bypass "from the back" of Σ, the result will be the mirror image of Figure 3. To study the effect of a bypass attachment on a torus, we first need to review the Farey graph. Given two rational numbers / and / , we define their Farey sum to be We also define their Farey multiplication to be Take the Poincaré disk in ℝ 2 and label the points (0, 1) as 0 = 0/1 and (0, −1) as ∞ = 1/0. Take the half circle with non-negative -coordinate. Pick a point in a halfway between two labeled points and label it with the Farey sum of the two points and connect it to both points by a geodesic. Repeat this process until all the positive rational numbers are a label on some point on the unit disk. Repeat the same for the half circle with non-positive -coordinate (for ∞, use the fraction −1/0). We call this disk with the labels the Farey graph, see Figure 4. Also notice that two rational numbers and satisfy | • | = 1 if and only if there is an edge between them in the Farey graph.
Consider a convex torus with the dividing set Γ , consisting of two homologically essential closed curves. Let be an attaching arc of a bypass for . Honda [33] completely studied what happens when is a part of a ruling curve for . Theorem 2.1 (Honda [33]). Suppose a convex torus has two dividing curves of slope , and is an attaching arc of a bypass for , which is a part of a ruling curve of slope . Let be the convex torus obtained from by attaching a bypass along . Then the dividing set Γ consists of two dividing curves of slope , where • if the bypass is attached from the front, then is the farthest point on the Farey graph clockwise of and counterclockwise of that is connected to by an edge (and if and are connected by an edge, then = ), • if the bypass is attached from the back, then is the farthest point on the Farey graph counterclockwise of and clockwise of that is connected to by an edge (and if and are connected by an edge, then = ).
In general, we can find a bypass lying on a convex surface if there exist boundaryparallel dividing curves on the surface. In particular, we can find a bypass in a convex disk with more than one dividing curves by applying the Legendrian realization principle.
Theorem 2.2 (Honda [33]). Let Σ be a convex surface and be a convex disk with Legendrian boundary. Suppose Σ and intersect transversely and Σ ∩ = . Suppose ( ) < −1. Then for any boundary-parallel dividing curve on , there exists a bypass for Σ containing .

Bypasses and contact isotopy.
We continue to review the properties of bypasses. Let Σ be a convex surface and be a bypass for Σ. Suppose the attaching arc of passes three dividing curves 1 , 2 and 3 consecutively. We say the bypass is effective if 2 is different from 1 and 3 . Honda showed [33] attaching an effective bypass to a torus will decrease the number of dividing curves if 1 , 2 and 3 are all different, or change the dividing slope of if 1 and 3 are the same (Theorem 2.1).
See Figure 5 for the first two cases. Recall that Giroux [26] proved that an -invariant neighborhood of a convex surface Σ is tight if and only if Σ 2 and there is no closed contractible dividing curve on Σ, or Σ 2 and there is a single dividing curve on Σ.
Thus the second type of bypasses does not occur in a tight contact structure. If a bypass does not change the dividing set, we call it a trivial bypass. Honda [35] showed that a trivial bypass is indeed trivial. [35]). Suppose Σ is a convex surface which is closed or compact with Legendrian boundary. If is a trivial bypass for Σ, then a neighborhood (Σ ∪ ), which is a result of the bypass attachment, is an -invariant neighborhood of Σ.

Lemma 2.3 (Honda
A rotative layer is a universally tight contact structure on 2 × with convex boundary such that the dividing slopes of 2 ×{0} and 2 ×{1} are different. Also, a non-rotative layer is a tight contact structure on 2 × [0, 1] with convex boundary such that any convex tori parallel to the boundary have the same dividing slope. Non-rotative layers were studied in [34,37]. One useful property is the attach=dig principle. We introduce a version of the principle for a simple case.  the dividing slope and the number of dividing curves on , respectively. Suppose 0 < 2 < 4 . Then after contact isotopy relative to 2 and the boundary, 2 × [1, 3] becomes an -invariant neighborhood with 1 = 2 = 3 and 1 = 3 = 2. Also, 1 can be obtained by attaching a sequence of bypasses from the back of 2 , and 3 can be obtained by attaching a sequence of bypasses from the front of 2 . Let Σ be a closed surface or a compact surface with boundary. Giroux [27] showed that we can perturb a contact structure on Σ × [0, 1] so that Σ × { } are convex for all but finite ∈ [0, 1], and a neighborhood of the non-convex Σ × { } is contactomorphic to a bypass attachment. After that, Honda and Huang [36] generalized it to every dimension. Theorem 2.5 (Giroux [27], Honda-Huang [36]). Let be a contact structure on Σ × [0, 1] such that Σ × {0} and Σ × {1} are convex. Then up to contact isotopy relative to the boundary, there exists a finite sequence 0 < 1 < · · · < < 1 such that • There exists > 0 for each such that Σ × [ − , + ] is contactomorphic to a bypass attachment.
We end this section by showing that if a contactomorphism fixes a convex surface or a Legendrian knot, then after contact isotopy, the contactomorphism also fixes a neighborhood of them. The second statement of Lemma 2.7 was proved in [10, Lemma 6], but we present a proof for completeness. (1) if = Σ is a compact convex surface, then there exist an -invariant neighborhood of Σ and a contactomorphism of ( , ) such that | = and is contact isotopic to .
= is a Legendrian knot, then there exist a standard neighborhood of and a contactomorphism of ( , ) such that | = and is contact isotopic to .
Proof. Consider the case = Σ first. Take  Notice that 1 is well-defined on ( ) but we can extend it to entire Σ × ℝ by extending the corresponding contact Hamiltonian. Now we define := 0 + (1 − ) 1 . Then for every ∈ [0, 1], the vector field is also a contact vector field since Let be the flow of . Since 0 = , we have Also, since 1 = * ( 0 ) on and | Σ = , we have for ( , ) ∈ . Define an isotopy of contact embeddings ( , ) := 1− ( , 0) for ( , ) ∈ and it is our desired isotopy. Now consider the case = . Take a standard neighborhood of which is contac- where ∼ + 1 is the coordinate on 1 = ℝ/ℤ, the pair ( , ) is the coordinates on ℝ 2 and is identified with 1 × {0}. Take another standard neighborhood = 1 × 2 where 2 is a small disk containing the origin such that ( ) ⊂ 1 × ℝ 2 . We will follow the same strategy as in the case of = Σ. That is, it is enough to find an isotopy of contact embeddings : ( , | ) → ( , ) satisfying 0 = | and 1 = .
We can write | in the form where : → 1 , : → ℝ and : → ℝ. In the local coordinates, we can rewrite the condition * ( ) = where : → ℝ + for | to be a contact embedding as follows: which is equivalent to Since | = , we have ). Thus we have an isotopy of contact embeddings Differentiate the first equation in (2) with respect to and we obtain Evaluate this equation at ( , 0, 0). Then by the equations in (3), we obtain Differentiate the third equation in (2) with respect to and we obtain Evaluate this equation at ( , 0, 0). Then by the equations in (3), we obtain Differentiate the first equation in (2) with respect to and we obtain Evaluate this equation at ( , 0, 0). Then by the equations in (3), we obtain Evaluate the equations in (2) at ( , 0, 0). Then by the equations in (3), we obtain Finally, from the equations (4), (5), (6) and (7), we obtain which is a contact embedding from ( , Then we can define another isotopy of contact embeddings as follows: Let be a concatenation of 1− and ℎ . This is our desired isotopy.

Tight contact structures on a solid torus and lens spaces.
Consider a tight contact structure on ( 1 , 2 ) = 2 × with a characteristic foliation ℱ on the boundary that is divided by two dividing curves of slope on 2 × { } for = 0, 1, where 0 and 1 are connected by an edge in the Farey graph. We say that a contact structure is minimally twisting if for any boundary-parallel convex torus in , the dividing slope is clockwise of 0 and counterclockwise of 1 in the Farey graph. Theorem 2.8 (Honda [33]). If ( 1 , 2 ) and ℱ are as above, then there exist exactly two minimally twisting tight contact structures on ( 1 , 2 ) that induce ℱ on the boundary, up to isotopy fixing ℱ .
The two contact structures given by Theorem 2.8 are distinguished by their relative Euler class. We call them positive and negative basic slices after picking an orientation. Let = 1 × 2 and choose coordinates for 1 ( ) such that 0 is a longitude 1 × { } (product framing), and ∞ is a meridian. Let / is a rational number and be the unique integer such that + ∈ [−1, 0), and Theorem 2.9 (Honda [33]). Suppose = 1 × 2 with two dividing curves of slope = / . Fix a characteristic foliation ℱ on that is divided by the dividing curves. Then there is one to one correspondence between tight contact structures on and ( , ), there is a unique tight contact structure and it is universally tight, (4) if ∉ ℤ, there are exactly two universally tight contact structures, (5) a tight contact structure on is universally tight if and only if it has the extremal relative Euler class, which evaluates to ±(| | − 1) on a convex meridian disk of whose boundary intersects the dividing curves on minimally. In this case, all dividing curves on is boundary-parallel. See Figure 9 for example.
To study tight contact structures on lens spaces, it is useful to use different coordinates for a solid torus. We say a tight contact solid torus with convex boundary is a solid torus with lower meridian if it has two dividing curves of slope with meridional slope , and any convex torus in the solid torus parallel to the boundary has a dividing slope clockwise of and counterclockwise of in the Farey graph. We denote it by ( , ; ). We say a tight contact solid torus with convex boundary is a solid tours with upper meridian if it has two dividing curves of slope with meridional slope , and any convex torus in the solid torus parallel to the boundary has a dividing slope counterclockwise of and clockwise of in the Farey graph. We denote it by ( , ; ). According to Theorem 2.9, both ( , ; ) and ( , ; ) admit a unique tight contact structure if and only if there is an edge between and in the Farey graph. We assume a solid torus 1 × 2 has lower meridian of slope ∞ unless otherwise specified.
We can also represent this decomposition on the Farey graph. Let 0 , . . . , be the shortest path from − / to 0 in the Farey graph clockwise of − / and counterclockwise of 0. Notice that 0 = − / , −1 = −1 and = 0. Decorate the edges in the path with + or − except for the first and the last ones. Each decorated edge from to +1 represents a basic slice ( , +1 ) and the decoration on the edge represents the sign of the basic slice. The first and the last edges represent the solid tori ( 1 , − / ; ) and (−1, 0; ), respectively. We can consider (−1, − / ; ) as a union of ( 1 , − / ; ) and ( 1 , −1). See Figure 6. For later usage, notice that Let ( , ) be a pair of coprime integers satisfying > > 0 and where ≤ −2 for 0 ≤ ≤ .
Theorem 2.10 (Giroux [27], Honda [33]). Suppose ( , ) are as above. Then There is another way to construct universally tight contact structures on lens spaces ( , ). Consider 3 as a unit sphere in ℂ 2 . Then the standard contact structure We can consider ( , ) as the quotient of 3 ⊂ ℂ 2 under the ℤ -action generated by Since is invariant under this ℤ -action, we obtain an induced contact structure on ( , ). We call this contact structure the standard contact structure on ( , ). However, one should notice that the standard contact structure is not unique in general. In fact, we can repeat the same construction on − and obtain another contact structure on ( , ). Although and − are isotopic in 3 , this is not the case for the induced contact structures on lens spaces in general. Thus we denote them by + and − , respectively. According to Theorem 2.10, + and − are isotopic if and only if ≡ −1 (mod ). Thus in this case, we just denote them by . Even in the case of −1 (mod ), since most of the arguments work in the same way, we will frequently denote them by and this means that we fix one of two standard contact structures. If 2 ≡ 1 (mod ), there exists an orientation preserving diffeomorphism on ( , ), which is defined by Also, there exists an orientation preserving diffeomorphism on any lens spaces ( , ), which is defined by We can check is a coorientation preserving contactomorphism of as follows: We can also check is a coorientation reversing contactomorphism of as follows: If ≡ −1 (mod ), since + and − are isotopic, we can apply the Moser's trick (see [21,Theorem 2.2.2]) and find an isotopy such that ( 1 ) * ( ± ) = ∓ . Thus 1 • is a coorientation preserving contactomorphism of which is smoothly isotopic to . We denote this by .
2.4. The mapping class group and rational unknots in lens spaces. The mapping class group of lens spaces was determined by Bonahon [3]. We warn the reader that his definition of lens spaces is different from ours. He defined ( , ) to be / -surgery on the unknot in 3 , while we defined ( , ) to be − / -surgery on the unknot in 3 (which is commonly used by contact topologists). Thus some statements below are different from the ones initially stated.
Lemma 2.14. The oriented rational unknots in ( , ) are given by up to smooth isotopy.
Proof. Bonahon [3] essentially showed that if an orientation preserving diffeomorphism : ( , ) → ( , ) sends 1 to 2 (up to isotopy), then it is smoothly isotopic to . He also showed that if sends 1 to − 1 , then it is smoothly isotopic to . Due to this fact, we only need to figure out when those diffeomorphisms are smoothly isotopic to the identity, which can be found in Theorem 2.13. If = 2, both and are smoothly isotopic to the identity. Thus all ± 1 and ± 2 are smoothly isotopic.
If ≡ −1, • is smoothly isotopic to the identity. Thus 1 is smoothly isotopic to − 2 , and − 1 is smoothly isotopic to 2 .
If ≡ 1, is smoothly isotopic to the identity, but is not. Thus 1 is smoothly isotopic to 2 , and − 1 is smoothly isotopic to − 2 .
If ±1, none of and is smoothly isotopic to the identity. Thus none of ± 1 and ± 2 is smoothly isotopic to each other.
Let ( , ) be a pair of coprime integers satisfying > > 0. Geiges and Onaran [23] depicted surgery presentations for the rational unknots 1 and 2 in ( , ), see Figure 1 and 7. Here, / is the largest rational number satisfying − = −1. We can write the negative continued fraction of − / as follows: where ≤ −2 for 0 ≤ ≤ . Then we have [40] for example). After taking the inverse of these matrices, we obtain Notice that the two surgery presentations in Figure 7 are not the same. However, by the equality (8), we can naturally identify one surgery presentation with the other as shown in Figure 8. Moreover, if we fix the signs of stabilization, then we can also identify one contact surgery presentation with the other in Figure 1. We end this section by reviewing the mapping class group of 1 × 2 and contactomorphisms of the standard contact structure on 1 × 2 . Consider 1 × 2 ⊂ 1 × ℝ 3 , where 2 is a unit sphere in ℝ 3 . Then the standard contact structure There exists an orientation preserving diffeomorphism of 1 × 2 which is defined by Consider a rotation matrix of ℝ 3 about -axis: There is another orientation preserving diffeomorphism of 1 × 2 , which is the Dehn twist about an essential sphere, defined by We can check is a coorientation preserving contactomorphism immediately: * ( ) = * ( + − ) Although is not a contactomorphism of , since there exists a unique tight contact structure on 1 × 2 , two contact structures and * ( ) are isotopic. We can apply the Moser's trick again and obtain an isotopy such that ( 1 ) * ( ) = * ( ). Then clearly −1 1 • is a contactomorphism of which is smoothly isotopic to . We just relabel −1 1 • as . The mapping class group of 1 × 2 was determined by Gluck [30]. After that, Hatcher [32] determined the homotopy type of Diff( 1 × 2 ). (Gluck [30], see also Hatcher [32]). The mapping class group of 1 × 2 is

Theorem 2.15
We define a positively oriented core of 1 × 2 to be and − to be its orientation reversal and call it a negatively oriented core. Notice that and − are not smoothly isotopic to each other. Chen, Ding and Li [8] classified Legendrian representatives of the oriented core in the standard contact structure on 1 × 2 up to Legendrian isotopy. Notice that the core is not (rationally) null-homologous, so the (rational) Thurston-Bennequin invariant is not well-defined. However, since ( ) = 0, the contact structure is trivial as a plane field, so the rotation number is well-defined. Ding and Geiges [10] also studied the effect of and on the rotation number of Legendrian representatives of the oriented cores in 1 × 2 . ). Let be a Legendrian representative of the positively oriented core in the standard contact structure on 1 × 2 and − be its orientation reversal. Then we have rot( (± )) = rot(± ) ± 1.

Invariants of rationally null-homologous Legendrian and transverse knots.
The classical invariants for null-homologous Legendrian and transverse knots were extended to rationally null-homologous knots by Baker and Etnyre [1]. Let be a Legendrian representative of a rationally null-homologous knot in a contact 3-manifold ( , ). Suppose the order of in 1 ( ) is and a rational Seifert surface for is Σ. Let be a push-off of along the contact framing. Then the rational Thurston-Bennequin invariant of is defined by Notice that there is an inclusion map : Σ → which is an embedding in the interior of Σ, and an -fold cover of on Σ. Since the pullback contact structure * ( ) is trivial as a plane field, the pullback of a non-vanishing tangent vector field * ( ) of gives a section on ℝ 2 × Σ after fixing a trivialization. This induces a Gauss map : 1 → 1 . We define the rational rotation number of as follows: Let be a transverse representative of a rationally null-homologous knot in a contact 3-manifold ( , ), and be a Legendrian representative such that is a (positive) transverse push-off of . Then the rational self-linking number of is defined by We also denote the maximum rational Thurston-Bennequin invariant among the Legendrian representatives of by tb ℚ ( ). Similarly, we can define the maximum rational self-linking number sl ℚ ( ). Baker and Etnyre [1] also showed that the stabilization has the same effect on the invariants as in the null-homologous case: Using these invariants, Baker and Etnyre [1] coarsely classified Legendrian rational unknots in any tight contact structure on lens spaces. where / is the largest rational number satisfying − = −1. Also the rational rotation numbers are determined by the formula in Lemma 2.19 or 2.20. For any Legendrian representative of rational unknots in , there is a contactomorphism of which is smoothly isotopic to the identity such that ( ) is one of the Legendrian representatives above, or their stabilization.
Proof. Recall from Section 2.3 and 2.4 that any tight contact structure on ( , ) can be decomposed into (−1, − / ; ) and (−1, 0; ), and 1 is the core of (−1, 0; ) and 2 is the core (−1, / ; ). Also, we showed that there is a Legendrian representative 1 of 1 , whose standard neighborhood is (−1, 0; ), see Figure 1. By Theorem 2.10, the decomposition is unique, so there exists a contactomorphism from one decomposition to another. Thus there exists unique 1 up to contactomorphism. Let be a Legendrian representative of 1 and be a standard neighborhood of . Notice that has longitudinal dividing curves, which implies there is an edge between the dividing slope and 0 in the Farey graph, so = 1/ for some ∈ ℤ. If ≥ 0, then a non-minimally twisting 2 × layer embeds in ( , ), which contradicts the tightness of . Thus ≤ −1 and is (1/ , 0; ) and the complement of is (1/ , − / ; ). If < −1, we can further decompose it into (−1, − / ; ) and (−1, 1/ ), a minimally twisting 2 × layer with slopes −1 and 1/ . Thus we can thicken using (−1, 1/ ) and obtain (−1, 0; ). Thus destabilizes to 1 . We can apply the same argument to − 1 and ± 2 . Now we are left to calculate the invariants. For the rotation numbers, see Lemma 2.19 or 2.20. Here, we calculate the Thurston-Bennequin invariants of ± for = 1, 2. We consider 1 first. Notice that the order of 1 is . Let be a convex Heegaard torus (−1, 0; ). Since the dividing slope of a standard neighborhood of 1 is −1, we can put a push-off of 1 along the contact framing on as a −1 slope curve (it is a Legendrian divide). Pick a meridian disk of (−1, − / ; ) for a rational Seifert surface of 1 . We can put the boundary of the meridian disk on as a − / slope curve. We can check the sign of each intersection point between these two curves is negative. Thus we have We use the second surgery presentation in Figure 1 for 2 . Then by the same argument above, we have There are two ways to calculate the rational rotation number of a Legendrian rational unknot: using contact surgery presentations by Geiges and Onaran [23], or using the Farey graph essentially due to Baker and Etnyre [1]. We introduce both methods.
It is well known that we can calculate the classical invariants from a contact surgery presentation for a given Legendrian knot in an integral homology sphere (see [12] for example). Geiges and Onaran [23] showed that the same formula works for contact surgery presentations for rationally null-homologous Legendrian knots in a homology sphere.
Consider a contact surgery presentation for a rationally null-homologous Legendrian knot in a homology sphere. Convert the contact surgery presentation into a (±1)- where lk( , ) is the linking number between and and rot 0 be the rotation number of in ( 3 , ). [23]). With the notations defined above, we have

Lemma 2.19 (Geiges-Onaran
Notice that if we change the orientation of , then rot 0 changes the sign and every component in lk also changes the sign while rot and do not change. Thus we have rot ℚ (− ) = − rot ℚ ( ). Now we introduce the second method. Recall from Section 2.3 that a decorated path for a tight contact structure on lens space ( , ) is the shortest path in the Farey graph from − / to 0, where all edges are decorated with + or − except for the first and the last ones. Let − / = 0 , 1 , . . . , = −1 be the vertices in . If −1 (mod ), we define where is the sign of the edge from to +1 . Here, we assume the numerator of is negative and the denominator of is positive. If ≡ −1 (mod ), then we define both 1 and 2 to be 0. Proof. Recall that 1 has the order in 1 ( ( , )) and its standard neighborhood is (−1, 0; ). Let = (−1, 0; ) and = (−1, − / ; ) that is the complement (−1, 0; ). Baker and Etnyre [1] showed that the rational rotation number of is equal to where is a non-vanishing section of | and is a meridian disk of . Decompose where ( , +1 ) is a basic slice with slopes and +1 . According to [33,Section 4.2], we can calculate the relative Euler class of a basic slice ( , +1 ) evaluated on a properly embedded annulus with − / slope boundary as follows: where is a non-vanishing section of restricted to ( , +1 ). Also, the relative Euler class of ( 1 , − / ; ) evaluates to 0 on a meridian disk by Theorem 2.9. Since the relative Euler class is additive under union, we obtain the formula in the statement by taking a summation. The same argument works for 2 .

LEGENDRIAN AND TRANSVERSE RATIONAL UNKNOTS IN LENS SPACES
In this section, we classify Legendrian and transverse rational unknots in any tight contact structure on lens spaces and prove the theorems in Section 1.2. To do so, we first determine the contact mapping class group of universally tight contact structures on a solid torus with two dividing curves. Then using this, we classify Legendrian representatives of the core in a tight contact structure on a solid torus with two dividing curves.
Before we start, we first extend the definitions. When ( , ) is a contact manifold with convex boundary, we define Cont( , ) = the group of contactomorphisms of ( , ) that are the identity on .
Also we define the contact mapping class group of ( , ) to be where ∼ if is contact isotopic to relative to the boundary. We start with determining the contact mapping class group of universally tight contact structures on 1 × 2 with two dividing curves. If the dividing curves are longitudinal, it was already determined by Giroux [28] and Vogel [41]. Proof. Let be a meridian disk of . After isotopy, we can assume that is convex and is Legendrian intersecting Γ minimally. According to Theorem 2.9, the relative Euler class of is extremal, which implies that every dividing curve on is boundary parallel (it is called a well-groomed dividing set). See Figure 9 for example. Consider a bypass whose attaching arc lies on . Notice that this bypass cannot be effective. Also, the attaching arc is one of the two configurations in Figure 5, and the bypass is trivial or yields a contractible dividing curve. Since is tight, the bypass must be trivial.
Let ∈ Cont( , ). After a small perturbation, we can assume that and ( ) intersect transversely in a finite set of circles. Choose an innermost circle among them. Then a disk in bounded by and a disk in ( ) bounded by form a sphere. Since is irreducible, this sphere bounds a ball, so using this we can isotope the disk in ( ) bounded by and reduce the number of intersection circles. See Figure 10  by attaching a bypass to . As we observed above, the only allowable bypasses for are trivial bypasses, so inductively all has the same dividing set and and +1 cobound an -invariant neighborhood by Lemma 2.3. Thus and +1 are contact isotopic for 1 ≤ ≤ − 1 and this implies that is contact isotopic to a contactomorphism fixing . By Lemma 2.7, we can further assume that fixes a small neighborhood of ∪ . Now pick a sphere contained in and parallel to a sphere \ . Perturb to be convex and let be the ball in bounded by . By Eliashberg [13, Thoerem 2.1.3], there exists a unique tight contact structure on up to isotopy fixing the characteristic foliation on . Also, according to Eliashberg [13,Theorem 2.4.2], we have 0 (Cont( , | )) = 1. This implies that | is contact isotopic to the identity relative to the boundary. Since | is the identity, is contact isotopic to the identity relative to the boundary and this completes the proof.
We need several steps to classify Legendrian and transverse rational unknots in tight contact structures on lens spaces. The first step is to classify Legendrian representatives of the core in a universally tight contact structure on a solid torus with two dividing curves. Legendrian knots in a solid torus with longitudinal dividing curves were already studied by Etnyre and Vértesi [15].

Proposition 3.2.
Let be a universally tight contact structure on a solid torus = 1 × 2 with two dividing curves of slope . Then the core of ( , ) is Legendrian simple: there exists a unique Legendrian representative with the maximum twisting number tw = where is the product framing of . Any Legendrian representative of the core is Legendrian isotopic to or its stabilization. Proof. We only consider the case ∈ [−1, 0) since we can realize any dividing slope by the Dehn twists about a meridian disk. We first show that there exists a unique Legendrian representative of the core of ( , ) with the maximum twisting number tw = −1 up to Legendrian isotopy. Suppose is a Legendrian representative of the core of ( , ). The dividing slope of a standard neighborhood of is an integer, so = −1 is the maximum twisting number. Let 1 and 2 be Legendrian representatives of the core with tw = −1. Suppose 1 and 2 are standard neighborhoods of 1 and 2 , respectively. Then we have (−1, ) = \ for = 1, 2, which are minimally twisting 2 × layers with the dividing slopes −1 and . According to Theorem 2.9, a tight contact structure on is completely determined by the tight contact structure on (−1, ). Thus there exists a coorientation preserving contactomorphism : 1 (−1, ) → 2 (−1, ) fixing . Since there exists a unique tight contact structure on a standard neighborhood of a Legendrian knot, we can extend to entire ( , ) so that ( 1 ) = 2 . By Theorem 3.1, is contact isotopic to the identity. Since sends 1 to 2 , they are Legendrian isotopic. Next, we will show that if is a Legendrian representative of the core of ( , ) with = tw ( ) < −1, then destabilizes. Suppose is a standard neighborhood of . Then we have ( , ) = \ , which is a minimally twisting 2 × layer with the dividing slopes and . Since < −1, we can decompose ( , ) into ( , + 1) ∪ ( + 1, ). Notice that ( , + 1) is a basic slice and we can thicken by attaching ( , + 1). This corresponds to a destabilization of .
Next, we improve the result by classifying the Legendrian representatives of the core in any tight contact structure on a solid torus with two dividing curves. Proposition 3.3. Let be a tight contact structure on a solid torus = 1 × 2 with two dividing curves of slope . Then the core of ( , ) is Legendrian simple: there exists a unique Legendrian representative with the maximum twisting number tw = where is the product framing of . Any Legendrian representative of the core is Legendrian isotopic to or its stabilization.
Proof. Again, we only consider the case ∈ [−1, 0) since we can realize any dividing slope by the Dehn twists about a meridian disk.
We first show that there exists a unique Legendrian representative of the core of ( , ) with the maximum twisting number tw = −1 up to Legendrian isotopy. Suppose is a Legendrian representative of the core of ( , ). The dividing slope of a standard neighborhood of is an integer, so = −1 is the maximum twisting number. Let 1 and 2 be Legendrian representatives of the core with tw = −1. Take a meridian disk of intersecting 2 transversely once. Perturb to be convex with Legendrian boundary such that intersects Γ minimally and intersects 1 transversely. We will consider two cases according to the intersection number between 1 and .
First, we consider the case | ∩ 1 | = 1. Suppose 1 and 2 are standard neighborhoods of 1 and 2 , respectively. After perturbing and , we can assume that the ruling slope of is ∞ and there exists a ruling curve that lies on and it is the only intersection between and for = 1, 2. Since tb( 1 ) = tb( 2 ) = −1 by the equality (1), each 1 and 2 intersects a dividing curve on at two points. See Figure 11 for example. Choose the dividing curves 1 , . . . , on such that 1 intersects 1 , 2 intersects , and and +1 are adjacent. We claim that we can isotope 1 through Legendrian knots so that 1 intersects 2 and does not intersect any other dividing curve on . Take a solid torus such that contains 1 and intersects in a closed curve that contains 1 and intersects 1 and 2 at four points. See Figure 12 for example. Perturb to be convex and to be Legendrian. Let be the dividing slope of . By the equality (1), we have tb( ) = −2. Due to this fact, there are only three cases we need to consider for the dividing curves on . The first case is > −1. Let 2 be the number of dividing curves on and = / for | | > ≥ 1. Since the dividing set interleaves, | ∩ Γ | = | ∩ Γ |. Thus we have The equality holds if and only if intersects Γ minimally. Since −1 < ≤ ∈ [−1, 0), we have > 1 and this implies that = 1. Thus there are two dividing curves on . Notice that the disk ⊂ , bounded by , contains two boundary-parallel dividing curves as shown in Figure 12. Notice that these two dividing curves are a part of 1 and 2 , but we just relabel them as 1 and 2 . According to Theorem 2.2, we can take a bypass lying on containing the dividing curve 1 . Remove a bypass attachment of this bypass from . Then by Theorem 2.1, the resulting solid torus 1 has two dividing curves of slope 1 satisfying −1 ≤ 1 < , and the resulting meridian disk 1 contains the single dividing curve 2 . Perturb 1 = 1 to be Legendrian. Then by equality (1), we have tb( 1 ) = −1. Since the dividing set interleaves, we have which implies that 1 is an integer. Thus 1 = −1 and 1 has two dividing curves of slope −1. Let 1 be a Legendrian representative of the core of 1 with tw = −1. Then 1 is a standard neighborhood of 1 . Since only contains boundary-parallel dividing curves, the restricted contact structure | is universally tight by Theorem 2.9. Since contains both 1 and 1 , by Proposition 3.2, 1 is Legendrian isotopic to 1 . Notice that 1 intersects 2 and does not intersect any other dividing curve on . The second case is = −1 and there are four dividing curves on . Again, the disk ⊂ , bounded by , contains two boundary parallel dividing curves as shown in Figure 12. According to Theorem 2.2, we can take a bypass lying on containing the dividing curve 1 . Remove a bypass attachment of this bypass from and let 1 be the resulting solid torus and 1 be the resulting meridian disk. Perturb 1 = 1 to be Legendrian. Since 1 contains the single dividing curve 2 , we have tb( 1 ) = −1. Thus there are two dividing curves on 1 as discussed in the first case. Since there are more than two dividing curves on , the bypass attachment does not change the dividing slope. Thus 1 has two dividing curves of slope −1 and \ 1 is a non-rotative layer. Let 1 be a Legendrian representative of the core of 1 with tw = −1. Then 1 is a standard neighborhood of 1 . By the attach=dig principle (Theorem 2.4), there is a solid torus containing with two dividing curves of slope −1. By Theorem 2.9, the restricted contact structure | is universally tight. Since contains both 1 and 1 , by Proposition 3.2, 1 is Legendrian isotopic to 1 . Notice that 1 intersects 2 and does not intersect any other dividing curve on .
The third case is = −1 and there are two dividing curves on . In this case, does not intersect Γ minimally. However, the disk ⊂ , bounded by , still contains two boundary parallel dividing curves as shown in Figure 12. According to Theorem 2.2, we can take a bypass lying on containing the dividing curve 1 . Remove a bypass attachment of this bypass from and let 1 be the resulting solid torus and 1 be the resulting meridian disk. Perturb 1 = 1 to be Legendrian. Since 1 contains the single dividing curve 2 , we have tb( 1 ) = −1. Thus there are two dividing curves on 1 as discussed in the first case. Since does not intersect Γ minimally, the bypass is not effective and the bypass attachment does not change the dividing slope. Thus 1 has two dividing curves of slope −1. Let 1 be a Legendrian representative of the core of 1 with tw = −1. Then 1 is a standard neighborhood of 1 . By Theorem 2.9, the restricted contact structure | is universally tight. Since contains both 1 and 1 , by Proposition 3.2, 1 is Legendrian isotopic to 1 . Notice that 1 intersects 2 and does not intersect any other dividing curve on .
We just have proved the claim. By applying the claim inductively, we can isotope 1 through Legendrian knots until 1 intersects and does not intersect any other dividing curve on . After that, take a bypass lying on which does not contain , and remove a bypass attachment of the bypass from . Repeat this until there is only one dividing curve, , left. See Figure 13 for example. Let be the resulting solid torus and be the resulting meridian disk. Perturb = to be Legendrian. Since contains the single dividing curve , we have tb( ) = −1. Let = / be the dividing slope of and 2 be the number of dividing curves. Since the dividing set interleaves, we have The equality holds if and only if intersects Γ minimally. From the inequality, we have = 1 and = 1. Thus there are two dividing curves on and is an integer. Since none of the bypasses does not intersect both 1 and 2 and the bypass attachment is a local operation, contains both 1 and 2 . Thus we have −1 ≤ ≤ ∈ [−1, 0) and this implies that = −1. Thus has two dividing curves of slope −1. By Theorem 2.9, the restricted contact structure is universally tight. Since contains both 1 and 2 , by Proposition 3.2, 1 and 2 are Legendrian isotopic. Next, we consider the case := | ∩ 1 | > 1. In this case, we can perturb 1 so that the ruling slope is ∞ and there are ruling curves 1 1 , . . . , 1 lying on and each 1 intersects a dividing curve on at two points. Similarly, we can also perturb 2 so that the ruling slope is ∞ and there is a ruling curve 2 lying on intersecting a dividing curve on at two points. Choose the dividing curves 1 , . . . , on such that 1 1 intersects 1 , 2 intersects , and and +1 are adjacent. We claim that we can isotope 1 through Legendrian knots so that 1 1 intersects 2 , while fixing other 1 for 2 ≤ ≤ . After perturbing , we can take a solid torus such that contains 1 and there are ruling curves 1 , . . . , of lying on such that = 1 for 2 ≤ ≤ , 1 contains 1 1 and 1 intersects 1 and 2 at four points. By the equality (1), we have tb( 2 ) = −1 and this implies that there are two dividing curves on and the dividing slope is an integer as discussed above. Also, since contains 1 , we have −1 ≤ ≤ ∈ [−1, 0) and = −1. Thus has two dividing curves of slope −1. This implies that 1 does not intersect Γ minimally. Since there are two boundary-parallel dividing curves on the disk ⊂ , bounded by 1 , we can find a bypass lying on that contains 1 according to Theorem 2.2. Remove a bypass attachment of this bypass from and let 1 be the resulting solid torus and 1 be the resulting meridian disk. Since 1 contains the single dividing curve 2 , there are still two dividing curves on 1 . Since 1 does not intersect Γ minimally, the bypass is not effective and the bypass attachment does not change the dividing slope. Thus 1 has two dividing curves of slope −1. Let 1 be a Legendrian representative of the core of 1 with tw = −1. Then 1 is a standard neighborhood of 1 . By Theorem 2.9, the restricted contact structure | is universally tight. Since contains both 1 and 1 , by Proposition 3.2, 1 is Legendrian isotopic to 1 . This completes the claim.
By applying the claim inductively, we can isotope 1 through Legendrian knots until 1 1 intersects while fixing other 1 for 2 ≤ ≤ . After that, apply the claim to 1 2 and we can isotope 1 through Legendrian knots until 1 2 intersects while fixing other 1 . Repeat the argument until all 1 for 1 ≤ ≤ intersect . Now using Theorem 2.2, take a bypass lying on that does not contain and remove a bypass attachment of this bypass from . Repeat this until there is only one dividing curve, , left. Let be the resulting solid torus and be the resulting meridian disk. Perturb = to be Legendrian. Since contains the single dividing curve , we have tb( ) = −1 and this implies that there are two dividing curves on and the dividing slope is an integer as discussed above. Since none of the bypasses does not intersect both 1 and 2 and the bypass attachment is a local operation, contains both 1 and 2 . Thus we have −1 ≤ ≤ ∈ [−1, 0) and this implies that = −1. Thus has two dividing curves of slope −1. By Theorem 2.9, the restricted contact structure is universally tight. Since contains both 1 and 2 , by Proposition 3.2, 1 and 2 are Legendrian isotopic.
Lastly, we show that if is a Legendrian representative of the core with := tw < −1, then destabilizes. Suppose is a standard neighborhood of . Then we have \ = ( , ), which is a minimally twisting 2 × layer with the dividing slopes and . Since < −1, we can decompose ( , ) into ( , + 1) ∪ ( + 1, ). Notice that ( , + 1) is a basic slice and we can thicken by attaching ( , + 1). This corresponds to a destabilization of . Now we are ready to classify Legendrian and transverse rational unknots in any tight contact structure on lens spaces. We first show the Legendrian simplicity. Let and be Legendrian representatives of with tb ℚ ( ) = tb ℚ ( ) = tb ℚ ( ), and and be standard neighborhoods of and , respectively. As shown in the proof of Theorem 2.18, both and are (−1, 0; ), i.e., they have two dividing curves of slope −1 with the upper meridional slope 0. Since and are smoothly isotopic, there exists a smooth isotopy from to . Then by Theorem 2.6, there exists a sequence of solid tori 1 , . . . , where 1 = , = and +1 is obtained by attaching a bypass to . Let be the dividing slope of . Here, we define Legendrian representatives associated to as follows. First, if ≤ −1, then contains a solid torus (−1, 0; ). Define to be a Legendrian representative of the core of this (−1, 0; ) with the maximum twisting number. Notice that this (−1, 0; ) is a standard neighborhood of so tb ℚ ( ) = tb ℚ ( ). If > −1, then is contained in some (−1, 0; ). Define to be a Legendrian representative of the core of this (−1, 0; ) with the maximum twisting number. Notice that this (−1, 0; ) is a standard neighborhood of so tb ℚ ( ) = tb ℚ ( ). From the definition, we can choose 0 to be and to be . We claim that and +1 are Legendrian isotopic, and this implies that and are Legendrian isotopic by induction. Observe that if > −1, then +1 ≥ −1 by Theorem 2.1. Similarly, if < −1, then +1 ≤ −1. Due to this fact, there are only two cases we need to consider. The first case is , +1 ≤ −1. Assume that +1 is obtained by attaching a bypass to which is contained in . In this case, contains +1 and this implies that contains both and +1 . If has more than two dividing curves, then by the attach=dig principle (Theorem 2.4), we can thicken and reduce the number of dividing curves. Also notice that and +1 still have the maximum twisting number in . If not, they destabilize and it contradicts that they have tb ℚ ( ). Thus by Proposition 3.3, and +1 are Legendrian isotopic. Now assume that +1 is obtained by attaching a bypass to which is not contained in . Then +1 contains and this implies that +1 contains both and +1 . Thus we can apply the same argument above (by switching the role of and +1 ) and conclude that and +1 are Legendrian isotopic. The second case is , +1 ≥ −1. In this case, the complements of each and +1 contains ( 1 , − / ; ) since 1 ≤ −1. Let and +1 be the Legendrian representatives of the core of each ( 1 , − / ; ) containing in and +1 , respectively, with the maximum twisting number. Notice that standard neighborhoods of and +1 are ( 1 , − / ; ). Assume that +1 is obtained by attaching a bypass to which is contained in . Then contains +1 and this implies that the complement of +1 contains both and +1 . Also notice that and +1 still have the maximum twisting number in the complement of +1 . If not, they destabilize and standard neighborhoods of them are ( , − / ; ) where is clockwise of 1 and there is an edge between and − / in the Farey graph. This implies that < − / or = ∞, and a non-minimally twisting 2 × layer embeds in ( ( , ), ), which contradicts the tightness of . Now and +1 are Legendrian isotopic by Proposition 3.3. Thus after Legendrian isotopy, we can identify with +1 , and then both and +1 are contained in the complement of a standard neighborhood of . By Proposition 3.3 again, and +1 are Legendrian isotopic. Now assume that +1 is obtained by attaching a bypass to which is not contained in . Then +1 contains and this implies that the complement of contains both and +1 . Thus we can apply the same argument above (by switching the role of and +1 ) and conclude that and +1 are Legendrian isotopic. This completes the claim.
By Theorem 2.18, a Legendrian representative of with tb ℚ ( ) < tb ℚ ( ) destabilizes. The identical argument works for − 1 and ± 2 . We leave them as exercises for the reader.
Remark 3.5. Notice that in the proof of Proposition 3.4, we need Proposition 3.3, not just Proposition 3.2 even for the universally tight contact structures on ( , ). This is because there exist virtually overtwisted neighborhoods of even in the universally tight contact structures.
Proof of Theorem 1.5. The theorem immediately follows from Theorem 2.18 and Proposition 3.4.
Proof of Theorem 1.6. In [17, Proof of Theorem 2.10], Etnyre and Honda showed that the classification of transverse knots is equivalent to the classification of Legendrian knots up to negative stabilization. Thus the theorem immediately follows from Theorem 1.5.

THE CONTACT MAPPING CLASS GROUP OF THE STANDARD LENS SPACES
In this section, we use the results from the previous sections to prove Theorem 1.1, and Corollary 1.2 and 1.4. We also prove Theorem 1.3 using the results of Ding and Geiges [10], see Section 2.4.
Recall from Section 2.3 that if 2 ≡ 1 (mod ), then is a contactomorphism of the standard contact structure on ( , ). Also, if ≡ −1 (mod ), then is smoothly isotopic to a contactomorphism of on ( , ). If −1 (mod ), then there exist two standard contact structure ± on ( , ) and * sends one to the other. We first show that any contactomorphism ∈ Cont( ( , ), ) is contact isotopic to either , , or the identity.
Next suppose is smoothly isotopic to . Let := −1 • . Then is a contactomorphism of which is smoothly isotopic to the identity. By the argument above, is contact isotopic to the identity. Thus and are contact isotopic.
Suppose is smoothly isotopic to . If ≡ −1 (mod ), then is smoothly isotopic to a contactomorphism as discussed above. Let := −1 • . Then is a contactomorphism which is smoothly isotopic to the identity. By the argument above, is contact isotopic to the identity, so and are contact isotopic. If −1 (mod ), then is a coorientation reversing contactomorphism sending ± to ∓ . Since and are isotopic, * ( ± ) is isotopic to * ( ± ) = ∓ , which contradicts that is a contactomorphism of ± . Now assume = 2. Then 0 (Diff + ( ( , ))) is trivial by Theorem 2.11. In this case, any contactomorphism is contact isotopic to the identity, so we have 0 (Cont( ( , ), )) = 1.
This completes the proof.
Proof of Corollary 1.2. In the proof of Theorem 1.1, we showed that any contactomorphism ∈ Cont( ( , ), ) is contact isotopic to either , , or the identity. This proves the injectivity of * since 0 (Diff + ( ( , ))) is generated by and .
Also, in the proof of Theorem 1.1, we showed that any diffeomorphism which is smoothly isotopic to cannot be a contactomorphism of when −1 (mod ). Moreover, is not isotopic to the identity if ≠ 2 by Theorem 2.13. This implies that * is not surjective when −1 (mod ). Finally, in the proof of Theorem 1.1, we showed that both 0 (Cont( ( , ), )) and 0 (Diff + ( ( , ))) are generated by if ≡ −1, so * is surjective.
Proof of Theorem 1.3. Suppose is a Legendrian representative of the positively oriented core of 1 × 2 with rot( ) = 0. First, observe that is not contact isotopic to if ≠ since rot( ( )) ≠ rot( ( )) by Lemma 2.17. Let be a contactomorphism of on 1 × 2 . We will show that and commute, and is contact isotopic to • for some ∈ ℤ and ∈ ℤ 2 . Then a map is a well-defined homomorphism and clearly it is an isomorphism. Suppose := rot( ( )).
There are two cases we need to consider according to the orientation of ( ). Suppose ( ) is smoothly isotopic to . According to Lemma 2.17, increases the rotation number of ( ) by 1, so we have rot(( − • )( )) = 0.
Thus ( − • )( ) is Legendrian isotopic to by Theorem 2.16. Moreover, by Lemma 2.7, we can assume that − • fixes a standard neighborhood of . Ding and Geiges [10] showed that the complement of in the standard contact structure on 1 × 2 is a solid torus with two longitudinal dividing curves. Now by Theorem 3.1, the restriction of − • to the complement of is contact isotopic to the identity. Thus − • is contact isotopic to the identity and hence is contact isotopic to . Now suppose ( ) is smoothly isotopic to − .