Planar Legendrian graphs
Abstract.
We classify topologically trivial Legendrian graphs and identify the complete family of nondestabilizeable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of a graph or as a subgraph will have an infinite number of distinct, topologically trivial nondestabilizeable Legendrian embeddings. Additionally, we introduce two moves, vertex stabilization and vertex twist, that change the Legendrian type within a smooth isotopy class.
Key words and phrases:
Contact Topology, Legendrian graphs2010 Mathematics Subject Classification:
53D10; 57M15; 05C101. Introduction
A Legendrian graph in a contact manifold is an embedding of an abstract graph such that the image of each edge is tangent to the contact structure. In contrast with the study of Legendrian knots and links, relatively little is known about Legendrian graphs. Only a few spatial graph types have been classified and there are several realization and nonrealization results for Legendrian graphs in or equivalently .
Eliashberg and Fraser proved that all Legendrian embeddings of an abstract tree are Legendrian isotopic [EF09]. The second author and Pavelescu classified topologically trivial Legendrian embeddings of the Lollipop and Handcuff graphs [OP12]. A graph has a Legendrian embedding with all cycles maximal unknots if and only if it does not contain as a minor [OP12]. However, does have a Legendrian embedding where each cycle has maximal ThurstonBennequin number in its smooth isotopy class [Tan15]. It is unknown whether every has such a Legendrian embedding. The second author and Pavelescu solved the geography problem for Legendrian embeddings of the graph where each cycle is unknotted [OP14]. They gave necessary and sufficient conditions for fixed vectors to be realized as ThurstonBennequin and rotation invariants of such Legendrian graphs. In [OP16] they found further restrictions on the ThurstonBennequin and rotation invariants of unknotted embeddings of complete and complete bipartite graphs.
In this paper, we completely classify topologically trivial Legendrian embeddings of the graph in . Our key technical tool is the main result in [LCO16] that the Legendrian ribbon and rotation invariant are a complete pair of invariants for topologically trivial Legendrian graphs.
Before stating the classification, we first describe some interesting consequences that distinguish Legendrian graphs from Legendrian knots. For a fixed isotopy class of embeddings, the set of Legendrian realizations up to Legendrian isotopy is “generated” by a set of nondestabilizeable Legendrian realizations. Specifically, there exists a set of Legendrian embeddings such that

every Legendrian embedding in the isotopy class is obtained by a finite sequence of stabilizations from some element of , and

no embedding in admits a destabilization.
See Section 2 for a precise description of stabilization. For all smooth knot types for which the set is known, it is finite. However, this is not true for topologically trivial Legendrian graphs. There exists an infinite family of nondestabilizeable realizations. The front projection of is given as follows. Fix two points in the plane and connect them by 3 arcs with no cusps. Now, take , collect the bottom 2 strands and add twists as in Figure LABEL:fig:Gl. Call this graph .
Theorem 1.1.
The infinite family , where and , are nondestabilizeable and pairwise nonLegendrian isotopic.
This result is a sharp contrast with what is known about Legendrian knots. Moreover, it is always true if is abstractly planar and contains a subdivision of the graph as a subgraph.
Theorem 1.2.
Let be an abstract planar graph that contains a subdivision of the graph or as a subgraph. Then there exists infinitely many, pairwisedistinct, topologically trivial Legendrian embeddings .
Fuchs and Tabachnikov [FT97] proved that if are Legendrian knots that are smoothly isotopic, then there exists a sequence of (de)stabilizations that connect and . However, the stabilization move (which we refer to here as an edge stabilization) is not sufficient to connect any two Legendrian realizations of the same spatial graph.
Theorem 1.3.
Two embeddings and are related by a sequence of edge stabilizations if and only if .
We introduce two new operations, called vertex stabilization and vertex twist, that modify Legendrian graphs without affecting the smooth isotopy class. We conjecture that, together with edge stabilizations, these form a complete set of operations relating any two Legendrian graphs in the same smooth isotopy class.
Conjecture 1.4.
Let be a graph and a smooth embedding. Then any two Legendrian realization and in of are related, up to Legendrian isotopy, by a sequence of edge stabilizations, vertex stabilizations, and vertex twists.
Using the techniques of [LCO16], we have a proof of this conjecture if is topologically trivial but not for arbitrary isotopy classes.
Now we address the classification. The graph consists of two vertices and three edges . There are three cycles . We orient the cycle so that is oriented from to . Throughout this paper, we let denote the abstract graph and denote a topologically trivial embedding . The ThurstonBennequin and rotation numbers of each cycle determine vectors
We also think of the pair as a vector in . Let denote the total rotation number of .
Recall that for the study of knotted graphs, there is an important distinction between embeddings and their images. Given an abstract graph , a spatial embedding , and an automorphism of , it is not true in general that there exists an ambient isotopy satisfying . Hence it is necessary to distinguish between embeddings of up to ambient isotopy and their images up to ambient isotopy. Pictorally, this corresponds to a difference between unlabeled diagrams up to equivalence and labeled diagrams up to equivalence. In particular, consider the graph in Figure 1. All possible ways to label the edges and vertices have the same image; however, there exists different labeling choices such that there is no sequence of Reidemeister moves that equates the two as labeled graphs. This is analagous to a noninvertible knot.
The automorphisms of are permutations of the vertices and edges and so the automorphism group is . Under , the image or orientation of a cycle may change and so the invariants change as well after relabeling by . In Section 3, we define representations of the symmetric groups on that describe the algebraic effect on the classical invariants of permuting the vertices and edges, respectively. Certain symmetries of under the action by correspond to relabelings of a graph.
Theorem 1.5.
The classification of topologically trivial Legendrian embeddings of the graph in is as follows:

The set for and is the set of nondestabilizeable, topologically trivial Legendrian embeddings .

If satisfy
Rot for , then there exists a topologically trivial Legendrian embedding with .

Let be an admissible pair. Then

if , there exist two distinct, topologically trivial embeddings with , and

if , there exists one topologically trivial embedding with .


Let be an admissible pair with . Then

if for some transposition , then up to relabeling there exists one topologically trivial embedding with .

if for every transposition , then up to relabeling there exists two topologically trivial embeddings with .

2. Background
2.1. Spatial graphs
An abstract graph or graph is a 1complex consisting of vertices and the edges connecting them. A spatial graph or spatial embedding is an embedding of a fixed abstract graph into a manifold . Two spatial graphs and are ambient isotopic if there exits an isotopy such that and As with knots, there is a set of Reidemeister moves for spatial graphs described by Kauffman in [Kau89].
There is an additional subtlety when working with spatial graphs. It is necessary to distinguish between embeddings of a graph up to ambient isotopy and its images up to ambient isotopy. Given an abstract graph , a spatial embedding , and an automorphism of , it is not true in general that there exists an ambient isotopy satisfying . In this article we are working with embeddings of graphs up to ambient isotopy. This corresponds to working with labeled diagrams up to equivalence. Without the labels the diagrams can look the same for different graphs because different edges are in the same positions. A spatial graph is topologically trivial if it lies on an embedded in .
2.2. Legendrian graphs
A (cooriented) contact structure on a 3manifold is a plane field where the 1form satisfies the nonintegrability condition . As a result, the contact structure induces an orientation on and the 2form orients the contact planes. The basic example is the standard contact structure on . This can be extended by the onepoint compactification of to the standard structure on .
A spatial graph is Legendrian if its image is tangent to the contact planes at every point. Two Legendrian graphs and are Legendrian isotopic if there exists a oneparameter family of Legendrian graphs , where . For a Legendrian graph in , the projection to the plane is called the front projection. Though we are working in we will work with a projection in ; this is possible since Legendrian graphs and isotopes of Legendrian graphs can be perturbed slightly to miss a chosen point. The front projection is usually given as an immersion, since at any crossing the strand that is over is determined by the slope of the tangents. However, at times we will show diagrams where the crossings are indicated for added clarity. A front projection of the Legendrian graph that is in general position with all double points away from vertices is called generic. Two generic front projections of a Legendrian graph are related by the three Reidemeister moves for knots and links together with two moves given by the mutual position of vertices and edges [BI09]. See Figure 2.
There are a number of different invariants for Legendrian graphs. The first two are a generalization of the classical invariants, Thurston–Bennequin, tb, and rotation number, rot, to graphs [OP12]. For a Legendrian graph , with an ordering on its cycles, the Thurston–Bennequin number, , is the ordered list of the Thurston–Bennequin numbers for its cycles. Similarly, the rotation number, , is the ordered list of the rotation numbers for its cycles.
In a front diagram, the ThurstonBennequin number and rotation number of an oriented cycle can be computed by the formulas
Next, a Legendrian ribbon for a Legendrian graph is a compact, oriented surface such that

contains as its 1skeleton,

has no negative tangencies to ,

there exists a vector field on tangent to the characteristic foliation of whose time flow satisfies ,

the oriented boundary of is positively transverse to the contact structure .
The Legendrian ribbon is unique up to ambient contact isotopy and thus an invariant of . This gives rise to a couple more invariants. The underlying unoriented surface is the contact framing of . The transverse link that forms the boundary of is the transverse pushoff of .
A topological knot type is Legendrian simple if are a complete set of invariants for Legendrian realizations of . In other words, there is at most one Legendrian realization of for each choice of . Several topological knot types are known to be Legendrian simple. A few examples are the unknot, the figure8 knot, and torus knots [EF09, EH01, DG07]. For Legendrian graphs much less is known about Legendrian simplicity. In the case of topologically trivial graphs, we have the following result:
Theorem 2.1 ([Lco16]).
Let be a trivalent, planar graph. Then is a complete set of invariants for topologically trivial Legendrian embeddings
For a Legendrian knot , different Legendrian knots in the same topological class can be obtained by stabilizations, where a single twist is added in an arc of the knot, consistent with the contact structure. This can be defined in with the front projection. A stabilization means replacing an arc in the front projection of by one of the zigzags in Figure 3. The stabilization is positive if the new cusps are oriented downwards and negative if the cusps are oriented upwards The classical invariants of the stabilized knot satisfy and . Thus stabilization changes the Legendrian isotopy class. It is a well defined operation and can be performed anywhere along a knot.
For Legendrian graphs, edge stabilizations along a fixed edge are defined in the same way. They can be performed at any point along a given edge but stabilizing along different edges will not be equivalent in general.
Consider the map diffeomorphism of to itself. It is an orientationpreserving contactomorphism of since . However, it reverses the coorientation of the contact planes. If is a Legendrian graph, the mirror is the image of under the map . In the front projection, this corresponds to mirroring across the xaxis.
Lemma 2.2.
Let be a Legendrian graph and let be its mirror. Then

the embeddings and are ambient isotopic,

the contact framings agree,

the Legendrian ribbons have opposite coorientations,

if is an oriented cycle of , then
Proof.
The map is a rotation by around the axis, so and are clearly smoothly isotopic. In addition, this rotation takes to but the coorientations on the contact framing induced by differ by a sign. Finally, the rotation does not change the number of cusps or the writhe of a cycle but it does exchange up and down cusps. Thus, is fixed but changes sign. ∎
3. Classification of graphs
The graph consists of two vertices and three edges . There are three cycles . We orient the cycle so that is oriented from to . Let be a Legendrian embedding. The ThurstonBennequin and rotation numbers of each cycle determine vectors
Let denote the total rotation number of . It is also convenient to think of the ThurstonBennequin invariant as a triple of halfintegers assigned to each edge. Define
for mod . It follows that .
Lemma 3.1.
Suppose that are topologically trivial Legendrian embeddings of such that . Then and are isotopic.
Proof.
Let be the Legendrian ribbons of and let be a smooth spheres containing the images of , respectively.
The graph is trivalent, so we can isotope and rel so that they agree in open neighborhoods of . Up to isotopy, the Legendrian ribbon can be obtained from a tubular neighborhood of in by adding half twists along the edge for . can be obtained from similarly. Since , this implies that the underlying unoriented surfaces and are isotopic. ∎
The contact planes and are oriented. Define the sign of a vertex to be
Lemma 3.2.
Let be a topologically trivial Legendrian embedding with rotation invariant . Then the total rotation number satisfies
For each Legendrian unknot , the ThurstonBennequin and rotation numbers must satisfy
(1)  
(2) 
Each cycle of a topologically trivial Legendrian must also satisfy these constraints. In addition, Lemma 3.2 implies that the total rotation number must satisfy
(3) 
A pair is admissible if it satisfies the above three restrictions.
Recall the set of graphs defined in Section 1. By abuse of notation, we let denote any embedding whose image is . Up to isotopy, there are possible labelings: there are ways to label the vertices and ways to label the edges.
Lemma 3.3.
Fix and let be an embedding with image . Then

if is the top edge, then

the total rotation number satisfies

is topologically trivial, and

is nondestabilizeable.
Proof.
To prove (1), note that pairing the top edge with either of the remaining edges gives a cycle with 2 cusps and no crossings. The cycle composed of the bottom two edges has 2 cusps and writhe . The edge twisting can now be immediately calculated from .
For (2), label the left vertex and right vertex and label the edges of at the left vertex in ascending order. This implies that . If , then the edges are labeled in ascending order at the right vertex and so as well. If , then . Thus, Lemma 3.2 implies the statement for this labeling of . Transposing the labels on the vertices or on a pair of vertices modifies by multiplication by and therefore does not change .
For (3), the initial graph is clearly topologically trivial. Moreover, twisting along the bottom strands does not change the ambient isotopy class. Finally, for (4), each is nondestabilizeable since each edge is contained in a cycle with . ∎
Lemma 3.4.
For each admissible , there exists a stabilization of some to an embedding such that .
Proof.
To simplify notation, define for . The vector determines a vector as described above. Up to cyclic relabeling of the edges, we can assume that . For , define
Admissibility implies that are nonnegative integers.
First, suppose that . Set and let be a nondestabilizeable Legendrian realization of with . As a result, by Lemma 3.3, and furthermore since . Apply positive stabilizations and negative stabilizations to the oriented edge for . A simple computation shows that the resulting graph satisfies and and is the required graph.
Secondly, suppose that . We claim that . There are four cases to prove this, depending on the value of .
If and , then
since . A similar argument holds if and .
If and , then
which implies that . So
In addition, and so . A similar argument holds if and .
Now, set and let such that . Apply positive and negative stabilizations to and then apply positive and negative stabilizations to for . Again, a simple computation shows that the resulting graph satisfies and and is the required graph. ∎
Lemma 3.5.
Let be admissible. If , then there exists two inequivalent Legendrian embeddings satisfying and .
Proof.
By Lemma 3.4, there exists some Legendrian embedding with and . It is obtained from some by performing positive and negative stabilizations along the edge for . See Figure 4. Let be the graph obtained by instead performing positive and negative stabilizations along the edge for . Consequently, and since , this also implies that . Let be the mirror of . Mirroring preserves and switches the sign of the rotation number of each cycle. Thus and .
We can perform these stabilizations away from the vertices, so we can assume that and agree near the vertices and thus have the same coorientation. In addition, since and both and are topologically trivial, there is a smooth isotopy of to taking to . The ribbon is smoothly isotopic to but has the opposite coorientation. Thus, and are distinguished by their Legendrian ribbons. ∎
Lemma 3.6.
Let be topologically trivial with the same and such that . Then and are Legendrian isotopic.
Proof.
Proposition 3.7.
Every topologically trivial Legendrian embedding is a stabilization of some .
Proof.
Corollary 3.8.
The set of nondestabilizeable, topologically trivial Legendrian graphs is precisely .
Theorem 2.1 combined with Lemmas 3.5 and 3.6 classify topologically trivial Legendrian embeddings of . We now address the classification of the images of these embeddings.
Relabeling the edges and vertices of the image corresponds to replacing with for some . The automorphisms of are permutations of the vertices and edges and so the automorphism group is . Under , the image or orientation of a cycle may change and so the invariants change as well after relabeling.
Let denote the automorphism that fixes the edges and swaps the vertices and let denote the automorphism that fixes the vertices and transposes the edges . The four automorphisms generate . The automorphism acts by reversing the orientation on cycles
and for , the automorphism acts on the set of oriented cycles by
To describe this effect on , we describe the induced representation of on . This representation is defined so that for every , the classical invariants satisfy
Let denote the trivial representation of on , i.e. is the identity for all . Let denote the sign representation of on , i.e. is multiplication by the sign of the permutation . Let denote the shifted permutation representation of on , i.e. if is an elementary transposition in swapping and , then transposes the coordinates . Define representations of , respectively, on by
and extend to a representation of . Identifying with and viewing as a vector in , we see that
From the above discussion, it follows that this is the correct effect of the classical invariants.
Proposition 3.9.
Suppose that are topologically trivial Legendrian graphs such that
for some . Then

If then are Legendrian isotopic up to relabeling.

If , then are Legendrian isotopic up to relabeling if and only if either

, or

there is a transposition such that
