A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface
Goldman and Turaev constructed a Lie bialgebra structure on the free -module generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket is zero if and only if is a power of a simple class. Chas constructed examples that show Turaev’s conjecture is, unfortunately, false. We define an operation in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through , so we can view as a generalization of . We show that Turaev’s conjecture holds when is replaced with . We also show that gives an explicit formula for the minimum number of self-intersection points of a loop in . The operation also satisfies identities similar to the co-Jacobi and coskew symmetry identities, so while is not a cobracket, behaves like a Lie cobracket for the Andersen-Mattes-Reshetikhin Poisson algebra.
We work in the smooth category. All manifolds and maps are assumed to be smooth unless stated otherwise, where smooth means .
Goldman  and Turaev  constructed a Lie bialgebra structure on the free -module generated by nontrivial free homotopy classes of loops on a surface . Turaev  conjectured that his cobracket is zero if and only if the class is a power of a simple class, where we say a free homotopy class is simple if it contains a simple representative. Chas  constructed examples showing that, unfortunately, Turaev’s conjecture is false on every surface of positive genus with boundary. In this paper, we show that Turaev’s conjecture is almost true. We define an operation in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams, and show that Turaev’s conjecture holds on all surfaces when one replaces with .
Turaev’s cobracket is a sum over the self-intersection points of a loop in a free homotopy class . Each term of the sum is a simple tensor of free homotopy classes loops, which are obtained by smoothing at the self-intersection point along its orientation. Each simple tensor is equipped with a sign coming from the intersection at (see Figure 1). Turaev’s conjecture is false because it is not uncommon for the same simple tensor of loops to appear twice in the sum , but with different signs.
We define the operation as a sum over the self-intersection points of a loop in , as in the definition of the Turaev cobracket. Rather than smoothing at each self-intersection point to obtain a simple tensor of two loops, we glue those loops together to create a wedge of two circles mapped to the surface. This can also be viewed as a chord diagram with one chord. As a result, terms of are less likely to cancel than terms of , and hence is less likely to be zero. In fact, Turaev’s conjecture holds when formulated for rather than :
Let be an oriented surface with or without boundary, which may or may not be compact. Let be a free homotopy class on . Then if and only if is a power of a simple class.
There is a simple relationship between and ; namely, if one smoothes each term of at the gluing point, and tensors the resulting loops, one obtains a term of (see Figure 1).
Hence the Turaev cobracket factors through , and we can view as a generalization of . The relationship between and is analogous to the relationship between the Andersen-Mattes-Reshetikhin Poisson bracket for chord diagrams and the Goldman Lie bracket. It is natural to wonder to what extent we can view as a cobracket for the Andersen-Mattes-Reshetikhin algebra. While is not a cobracket, in the final section of the paper, we show that satisfies identities similar to coskew symmetry and the co-Jacobi identity.
The operation also gives an explicit formula for the minimum number of self-intersection points of a generic loop in a given free homotopy class . We call this number the minimal self-intersection number of and denote it by . Both Turaev’s cobracket and the operation give lower bounds on the minimal self-intersection number of a given homotopy class . We call a free homotopy class primitive if it is not a power of another class in . Any class can be written as for some primitive class and . It follows easily from the definitions of and that is greater than or equal to plus half the number of terms in the (reduced) linear combinations or . More formally, the number of terms of a reduced linear combination of simple tensors of classes of loops, or of classes chord diagrams, is the sum of the absolute values of the coefficients of the classes. Chas’ counterexamples to Turaev’s conjecture show that the lower bound given by cannot, in general, be used to compute the minimal self-intersection number of . However the lower bound given by is always equal to :
Let be an oriented surface with or without boundary, which may or may not be compact. Let be a nontrivial free homotopy class on such that , where is primitive and . Then the minimal self-intersection number of is equal to plus the half number of terms of .
We briefly summarize some results related to Turaev’s conjecture and to computing the minimal self-intersection number. Le Donne  proved that Turaev’s conjecture is true for genus zero surfaces. For surfaces of positive genus, one might wonder to what extent Turaev’s conjecture is false. Chas and Krongold  approach this question by showing that, on surfaces with boundary, if and is at least a third power of a primitive class , then is simple.
A nice history of the problem of determining when a homotopy class is represented by a simple loop is given in Rivin . Birman and Series  give an explicit algorithm for detecting simple classes on surfaces with boundary. Cohen and Lustig  extend the work of Birman and Series to obtain an algorithm for computing the minimal intersection and self-intersection numbers of curves on surfaces with boundary, and Lustig  extends this to closed surfaces. We give an example which shows how one can algorithmically compute using on surfaces with boundary, though generally we do not emphasize algorithmic implications in this paper.
A different algebraic solution to the problem of computing the minimal intersection and self-intersection numbers of curves on a surface is given by Turaev and Viro . The advantage of is that it has a simple relationship to and pairs well with the Andersen-Mattes-Reshetikhin Poisson bracket. In fact, Chernov  uses the Andersen-Mattes-Reshetikhin bracket to compute the minimum number of intersection points of loops in given free homotopy classes.
2. The Goldman-Turaev and Andersen-Mattes-Reshetikhin Algebras and the Operation
2.1. The Goldman-Turaev Lie Bialgebra
We will now define the Goldman-Turaev Lie Bialgebra on the free -module generated by the set of free homotopy classes of loops on , which we denote by . Let , and let and be smooth, transverse representatives of and , respectively. We will use square brackets to denote the free homotopy class of a loop. The set of intersection points , or just when the choice of and is clear, is defined to be
Let denote the product of and as based loops in , where and . If is the image of more than one ordered pair in , then there is more than one homotopy class in corresponding to (or to ), so we choose a class as follows: Let be the induced map on the fundamental groups. Let be the generator of whose orientation agrees with the chosen orientation of . Then the class of in is given by . We choose the class of in in the same way. In particular, we must specify preimages of under and (i.e. a point in ) for the notation to make sense.
The Goldman bracket  is a linear map , defined by
where if the orientation given by the pair of vectors agrees with the orientation of , and otherwise. To check that the definition of is independent of the choices of and , one must show that does not change under elementary moves for a pair of smooth curves in general position. Using linearity, the definition of can be extended to all of .
Next we define the Turaev cobracket . Let be a free homotopy class on , and let be a smooth representative of with transverse self-intersection points. Let , or just when the choice of is clear, denote the set of self-intersection points of the loop . Let be the diagonal in . Elements of will be points in modulo the action of which interchanges the two coordinates. Now we define
Let be a self-intersection point of . Let denote the arc of going form to in the direction of the orientation of , and let denote the arc of going from to in the direction of the orientation of . Since , then and are loops. We assign these loops the names and in such a way that the ordered pair of tangent vectors gives the chosen orientation of . Now we let be the subset of which contains only self-intersection points such that the loops are nontrivial:
The Turaev cobracket is a linear map which is given on a single homotopy class by
One can show that the definition of is independent of the choice of by showing does not change under elementary moves for a smooth loop in general position. Using linearity, this definition of can be extended to all of .
Together, and equip with an involutive Lie Bialgebra structure [12, 18]. That is, and satisfy (co)skew-symmetry, the (co) Jacobi identity, a compatibility condition, and . A complete definition of a Lie Bialgebra is given in .
2.2. The Andersen-Mattes-Reshetikhin Algebra of Chord Diagrams
We now summarize the Andersen-Mattes-Reshetikhin algebra of chord diagrams on [1, 2]. A chord diagram is a disjoint union of oriented circles , called core circles, along with a collection of disjoint arcs , called chords, such that
1) for , and
A geometrical chord diagram on is a smooth map from a chord diagram to such that each chord in is mapped to a point. A chord diagram on is a homotopy class of a geometrical chord diagram , denoted .
Let denote the free -module generated by the set of chord diagrams on ( uses coefficients in , but we use here for consistency). Let be the submodule generated by a set of -relations, one of which is shown in Figure 3. The other relations can be obtained from this one as follows: one can reverse the direction of any arrow, and any time a chord intersects an arc whose orientation is reversed, the diagram is multiplied by a factor of -1.
Given two chord diagrams and on , we can form their disjoint union by choosing representatives (i.e., geometrical chord diagrams) of , taking a disjoint union of their underlying chord diagrams, mapping the result to as prescribed by the , and taking its free homotopy class. The disjoint union of chord diagrams defines a commutative multiplication on , giving an algebra structure with as an ideal. Let , and call this the algebra of chord diagrams.
Andersen, Mattes, and Reshetikhin [1, 2] constructed a Poisson bracket on , which can be viewed as a generalization of the Goldman bracket for chord diagrams on rather than free homotopy classes of loops. Let and be chord diagrams on , and choose representatives of . We define the set of intersection points , or just when the choice of and is clear, to be
where is a point in the preimage of the geometrical chord diagrams . For each with , let denote the geometrical chord diagram obtained by adding a chord between and . It is necessary to specify preimages of for this notation to be well-defined. Since each copy of in the chord diagram is oriented, we can define as before. The Andersen-Mattes-Reshetikhin Poisson bracket is defined by
where square brackets denote the free homotopy class of a geometrical chord diagram. This definition of can be extended to all of using bilinearity. For a proof that does not depend on the choices of , , see . In particular, it is necessary to check that is invariant under elementary moves, including the Reidemeister moves and the moves in Figures 4 and 5, and the -relations.
2.3. The Operation
The definition of given in this section is the simplest for the purposes of computing the minimal self-intersection number of a free homotopy class . In this section, we define only on free homotopy classes. In the final section of this paper, we modify the definition of in a way that allows us to more easily state an analogue of the co-Jacobi identity, and which allows us to extend the definition of to certain chord diagrams in the Andersen-Mattes-Reshetikhin algebra. The modified definition agrees with the definition below for free homotopy classes.
For this defintion of , we will need to use chord diagrams with oriented chords. Suppose is an oriented chord with its tail at and its head at in a geometrical chord diagram . We say agrees with the orientation of if the ordered pair of vectors gives the chosen orientation of . When we draw the image of a geometrical chord diagram, we label the image of a chord with a ’ if agrees with the orientation of , and we label it with a ‘’ otherwise.
Let denote the free -module generated by chord diagrams on consisting of one copy of , and one oriented chord connecting distinct points of that copy of . In addition to the usual Reidemeister moves, we have two additional elementary moves for diagrams with signed chords. These moves, with one possible choice of orientation on the branches, are shown in Figures 6 and 7, where denotes the sign on the chord. We define a linear map .
Let be a geometrical chord diagram on with one core circle. For each self-intersection point of , we let (respectively ) be the geometrical chord diagram obtained by adding an oriented chord between and that agrees (respectively, does not agree) with the orientation of .
Now we define on the class of the geometrical chord diagram by
Using linearity, we can extend this definition to all of . It remains to check that is independent of the choice of representative of .
2.4. is independent of the choice of representative of
We check that is invariant under the usual Reidemeister moves:
Regular isotopy: Invariance is clear.
First Reidemeister Move: This follows from the definition of .
Second Reidemeister Move: This follows from the move in Figure 6.
Third Reidemeister Move: This follows from the move in Figure 7.
We note that when checking invariance under the second and third moves, one must consider the case where some of the self-intersection points are in but not in .
2.5. Alternative notation for
We would like to show that factors through . To do this, we will rewrite the definition of for a free homotopy class in a way that makes its relationship to more transparent. Let and be loops in based at , such that and . We define a geometrical chord diagram which glues the loops and at the point . The underlying chord diagram of contains one core circle , and one oriented chord with its head at and its tail at . The geometrical chord diagram maps the chord to . Then we define and .
Now we are ready to rewrite the definition of for . Let be a representative of , and for each with , let and be the loops we defined for the Turaev cobracket. Now
2.6. Relationship between , the Goldman-Turaev Lie bialgebra, and the Andersen-Mattes-Reshetikhin Algebra of Chord Diagrams
Andersen, Mattes and Reshetikhin  show that there is a quotient algebra of which corresponds to Goldman’s algebra. Let be the ideal generated by the relation in Figure 8. In the quotient , each chord diagram is identified with the disjoint union of free homotopy classes obtained by smoothing the diagram at the intersections which are images of chords.
One can check that is a Poisson algebra homomorphism and is a Poisson algebra with an underlying Lie algebra that corresponds to Goldman’s algebra .
There is a similar relationship between the Turaev cobracket and . Let be the map which smoothes the chord diagram according to its orientation at an intersection which is an image of a chord, and tensors the two resulting homotopy classes together (see Figure 9). Then .
Remark: Turaev [18, p. 660] notes that the Turaev cobracket can be obtained algebraically from an operation defined in Supplement 2 of . It is possible that may be obtained from this operation as well. We do not know a way of obtaining Turaev’s operation from .
3. Proofs of Theorems
In this section, we prove Theorems 1.1 and 1.2. Recall that Theorem 1.1 states that if and only if is a power of a simple class. Theorem 1.2 gives an explicit formula for . We begin by describing two types of self-intersection points of a loop which is freely homotopic to a power of another loop. Then we prove Theorem 3.2, which describes when certain terms of cancel. Theorems 1.1 and 1.2 are corollaries of Theorem 3.2.
3.1. Intersection Points of Powers of Loops
Our goal is to understand the conditions under which different terms of cancel, when is a power of another class in . To do this, we need to distinguish between two different types of self-intersection points of a curve. Suppose we choose a geodesic representative of . Either all self-intersection points of are transverse, or has infinitely many self-intersection points, and in particular, is a power of another geodesic. Let be a point on the image of which is not a transverse self-intersection point of . Let be a geodesic loop such that in , and such that there is no geodesic such that (it is possible that ). Now we know that has finitely many self-intersection points, all of which are transverse. Let be the number of self-intersection points of . Since is orientable, we can perturb slightly to obtain a loop as follows: We begin to traverse beginning at , but whenever we are about to return to , we shift slightly to the left. After doing this times, we must return to and connect to the starting point. This requires crossing strands of the loop, creating self-intersection points. We call these Type 2 self-intersection points. For self-intersection point of , we get self-intersection points of (see Figure 10). We call these self-intersection points Type 1 self-intersection points. We note that we are counting self-intersections with multiplicity, as some of the self-intersection points of may be images of multiple points in .
Given a transverse self-intersection point of , we will denote the corresponding set of Type 1 self-intersection points of by , where is the label on the strand corresponding first branch of at (i.e., a strand going from top to bottom in Figure 11), and is the label on the strand corresponding to the second branch of at (i.e., a strand going from left to right in Figure 11). This relationship between the numbers of self-intersection points of and can be found in  for both orientable and non-orientable surfaces.
Let be a geodesic representative of , with , and and are as defined in the paragraph above. Then the contribution to of a Type 1 self-intersection point is
where , , and such that .
Proof. We will compute the contribution to for a Type 1 self-intersection point of , where is the perturbed version of described in the above paragraph. These terms are and . However, when we record the terms of , we perturb back to , so that the terms we record are geometrical chord diagrams whose images are contained in the image of and whose chords are mapped to . To compute , we begin at along the branch corresponding to , and wish to know how many times we traverse branches corresponding to and before returning to . The first time we return to , we must return along the branch of . Therefore for some integer . If we begin at along the branch corresponding to , we return to for the first time on the branch of . Therefore for some integer . But if we traverse followed by , we must traverse exactly once, so . ∎
3.2. Canceling terms of
Throughout this section, we will use the following facts, which hold for a compact surface with negative sectional curvature (though compactness is not needed for ).
Nontrivial abelian subgroups of are infinite cyclic.
There is a unique, maximal infinite cyclic group containing each nontrivial .
Two distinct geodesic arcs with common endpoints cannot be homotopic.
Each nontrivial contains a geodesic representative which is unique up to choice of parametrization.
The first fact holds by Preissman’s Theorem. The second fact is true if because is free. If is closed, the second fact follows from the proof of Preissman’s Theorem . The third and fourth facts can be found in , as Theorems 1.5.3 and 1.6.6 respectively.
We now show that for any free homotopy class on a compact surface, it is possible to choose a representative of such that no two terms coming from Type 1 intersection points cancel. This proof is based on ideas in  and . Later we will see that if or the annulus , geodesic loop on has no Type 1 self-intersection points, so in Theorem 3.2, we only consider surfaces of negative curvature.
Let be a compact surface equipped with a metric of negative curvature. Let . If is a geodesic representative of , then no two terms of corresponding to Type 1 intersection points of cancel.
Proof. Throughout this proof, denotes a free homotopy class (either of a geometrical chord diagram or a loop), denotes a homotopy class in , and denotes the homotopy class of a path from to with fixed endpoints. When we concatenate two paths and , we write , where the path written on the left is the path we traverse first.
We write for some geodesic loop and some , where is not a power of another loop. Suppose has self-intersection points, and let be a perturbation of with Type 1 self-intersection points and Type 2 self-intersection points. Let and be the sets of self-intersection points corresponding to the (transverse) self-intersection points and of respectively, with the indexing as defined in the previous section. We assume is nontrivial, since the theorem clearly holds when is trivial ( is in fact empty).
We wish to show that the terms of corresponding to points and cannot cancel. We suppose these terms cancel, and derive a contradiction.
First, we consider the case where for , but and may or may not be equal, and and may or may not be equal. In other words, and come from the same set of type 1 self-intersection points. Let and let . If either or , then by Lemma 3.1, the terms corresponding to and are
for integers such that . If and , then corresponds to a single element of , so we have just the first two of the above terms. In either case, it suffices to assume that the terms and cancel, where and may or may not be equal, and and may or may not be equal.
Then there exists such that
Conjugating Equation 3.3 by tells us that and commute, since . Similarly, conjugating Equation 3.4 by tells us and commute. Therefore the subgroups and are infinite cyclic, and are generated by elements and of , respectively. Note that these subgroups are nontrivial since is nontrivial. Fact states that each nontrivial element of is contained in a unique, maximal infinite cyclic group. Let and be the generators of the unique maximal infinite cyclic groups containing and respectively. Since is not freely homotopic to a power of another class, we have that and . But and are also infinite cyclic groups containing and , respectively. By the maximality of the , we have that and . This tells us and are powers of and respectively, so
The powers and can be either zero, positive, or negative. Once we make all possible cancellations in Equation 3.5, we will have two geodesic lassos (one on each side of the equation) formed by products of , , or their inverses, representing the same homotopy class in . Therefore these geodesic lassos must coincide. The geodesic on the left hand side of Equation 3.5 can begin by going along either or (depending on the sign of ), while the geodesic on the right hand side can begin along either or (depending on the sign of ). Therefore and must either be powers of the same loop, which is impossible, because we assumed is not a power of another class, or and must be trivial, which is impossible because of the definition of . Therefore the terms of corresponding to and cannot cancel when .
Now we will show that the terms of which correspond to and cannot cancel when and correspond to different ordered pairs in . Let , , , and . By Lemma 3.1 the terms which and contribute to are:
where . We will suppose that , and derive a contradiction. Switching the orders of the two loops on both sides of the equation gives us the equality , so if we assume that one of these equalities holds, all four terms above will cancel.
As in the case where , we will use the equality to find abelian subgroups of . To do this, we examine the Gauss diagram of with two oriented chords corresponding to the self-intersection points and . The four possible Gauss diagrams with two oriented chords are pictured in Figure 12.
We use the convention that each oriented chord points from the second branch of to the first branch of , where the branches of at a self-intersection point are ordered according to the orientation of . As shown in Figure 12, we let , , denote the arcs between the preimages of and . We let denote the image of the arc under .
We first change the basepoint of the first term from to , replacing by . Assuming the terms cancel, we can find such that
This allows us to rewrite Equations 3.8 and 3.9 just in terms of and the . Note that for diagrams and , we get the same two equations from 3.8 and 3.9, because the values of and , and the values of and , are interchanged. Therefore it suffices to consider diagrams , , and . The arguments for diagrams and are similar, so we will only examine and .
Diagram : In this case, , , , and (see Table 1). We express Equations 3.8 and 3.9 in terms of the to obtain
Therefore and commute, as do and . Since abelian subgroups of are infinite cyclic, the subgroups and are generated by elements and in respectively. Each nontrivial element of is contained in a unique, maximal infinite cyclic group by Fact . Let and be the generators of the unique maximal infinite cyclic groups containing and respectively. By assumption, is not freely homotopic to a power of another class. Therefore and . But and are also infinite cyclic groups containing and , respectively, so by the maximality of and , we have and . Thus and for some and . Now
so the path homotopy classes and are equal. Once we cancel with wherever possible, and