The degree of the colored HOMFLY polynomial
The colored HOMLFY polynomial is an important knot invariant depending on two variables and . We give bounds on the degree in both and generalizing Morton’s bounds [Mor86] for the ordinary HOMFLY polynomial. Our bounds suggest that the degree detects certain incompressible surfaces in the knot complement and perhaps more generally features of the character varieties of the knot group. We formulate a precise conjecture along these lines generalizing the slope conjecture of Garoufalidis [Gar11]. We prove our conjecture for all positive knots.
Our technique is a reformulation of the MOY state sum [MOY98] using -analogues of Ehrhart polynomials. As a direct application we explicitly compute the coefficients of -colored HOMFLY polynomial of any positive braid.
The colored HOMFLY polynomial is an important invariant in knot theory. It has many intriguing connections to string theory contact homology integrable systems Gromov-Witten invariants. Also much effort was directed to understanding its categorification. Given all this interest it is rather surprising that not much seems to be known about some of its simplest properties such as its degree. Even simple bounds on the maximum degree seem to be unavailable in the literature. The purpose of this paper is to provide such bounds and to show that the degree is actually an interesting invariant already.
The version of the colored HOMFLY polynomial we are discussing unifies all the Reshetikhin-Turaev quantum invariants of knots colored with symmetric powers of the standard representation. Our notation is for the unreduced -colored HOMFLY polynomial of where each component of is colored by the -th symmetric power. Setting yields the corresponding invariant. In particular we obtain the colored Jones polynomial for .
Despite its name is not exactly a two variable polynomial. It is rather a rational function in and . As such it still makes sense to talk about its maximal degree in . Indeed we can always expand in a Laurent series in . By definition this has a highest degree term in and this then is the maximal degree in , notation . The same goes for the variable .
Let be any oriented link diagram . The number of positive and negative crossings are denoted and the number of positively/negatively oriented Seifert circles is .
For positive diagrams
We get similar bounds for the the minimal degree by considering the mirror image of the diagram , because . These bounds also apply to the anti-symmetric version of the colored HOMFLY polynomial because of the formula [Zhu13].
These bounds should be compared to Morton’s bounds [Mor86] in the case . In our notation he proved that in perfect agreement with our bound. For the -degree Morton’s upper bound is . This is much sharper than our general -bound and matches our lower bound in the positive case. We conjecture there actually is equality for all positive diagrams.
Let us illustrate our bounds in the simple case of the closure of a -braid with crossings, [GNS15].
First the maxdegree in . All crossings are of the same sign so since . For any it is obtained we get terms of -degree attaining our upper bound since Next the . For we have a positive diagram with . Looking at the term we find the maximal degree of For our bound is very poor. In the formula the maxdegree is obtained by looking at the term. We find . This is about half of the given upper bound since .
1.1 Topological interpretation of the -degree
As an application for our bounds we give a topological interpretation of the growth rate of the -degree. For simplicity we restrict ourselves to the knot case, see [vdV13] for a similar conjecture in rank for links.
Recall the knot exterior is defined as , where is an open tubular neighborhood of the knot . is a compact -manifold with torus boundary and a canonical choice of a basis . A properly embedded essential surface has boundary slope if . Hatcher showed every knot has only finitely many slopes [Hat82]. For Montesinos knots there is an algorithm to compute these slopes. In particular it shows that any rational number can be the slope of some knot. Culler and Shalen showed that some slopes are the slopes of the Newton polygon of the A-polynomial of the knot.
For a knot define the set of HOMFLY slopes as follows
where means the set of limit points (accummulation points) of any set .
Also let denote the set of boundary slopes of essential surfaces as defined above.
Motivated by the AJ conjecture and the slope conjecture [Gar11] for the colored Jones polynomial, i.e. the specialization , we propose
(HOMFLY slope conjecture)
For any knot we have
Since positive knots are fibered and the fiber surface is essential [Sta78] with slope our degree bounds immediately imply and hence:
The HOMFLY slope conjecture is true for all positive knots.
Further evidence is provided by our negative -braid examples. We have seen that the . Therefore . For any alternating knot , see for example [Gar11] so the HOMFLY slope conjecture holds.
Comparing to the original slope conjecture it is likely that there are knots such that contains fractions with high denominators [GvdV14]. This gives an interesting constraint on any state sum for the colored HOMFLY polynomial: Either it involves a huge amount of cancellation or it explicitly encodes such fractions.
1.2 The head of the colored HOMFLY
An attractive way to organize the colored HOMFLY polynomial is as a Laurent series
This goes one step beyond our investigation of the maximal -degree. For positive diagrams and the leading term of such an expansion was called the top of the HOMFLY polynomial [KM13] (using the variable ).
The above expansion is useful because the coefficients seem to stabilize once scaled properly. Similar stabilization phenomena were discussed for positive and alternating knots in the colored Jones case [AD13, GL11]. Although unproven this stabilization is likely to persist for general knots after one passes to properly chosen arithmetic subsequences in .
As a working definition let us define the head of the colored HOMFLY as a rescaled version of the top coefficients.
The head of the colored HOMFLY polynomial is defined as
where is a monomial chosen such that the leading term of is .
The head is expected to be independent of in the following sense. Let be a power series such that the highest coefficients in are equal to the head as before. As mentioned this definition needs to be adjusted by passing to subsequences for general knots. For positive knots however it is fine.
For any positive braid closure, the head and equal to the head of the unknot, i.e. equal to the first coefficients of
The proof of this theorem shows the power of our new state sum in thinking about rather general situations in a diagrammatic fashion. The idea is to reduce to the case of -braids. Recall that for -braids we have an explicit formula (1). For the head is provided by the term only. It is then clear from the formula that the head is independent of and given by . Note that this family includes the unknot as .
The same formula for shows that the notion of head is not vacuous. In this case the head comes from the term only and can be expressed by the power series
1.3 Plan of the paper
In the first section we develop a brief theory of -Ehrhart polynomials that is perhaps interesting in its own right. The -Ehrhart polynomials play an essential role in our reformulation of the MOY state sum for the colored HOMFLY polynomial. The next two sections apply the state sum to derive our degree bounds and compute the head of the colored HOMFLY polymomial. Finally we prove our state sum by showing how it relates to the original MOY approach. We end with a discussion.
The author thanks Tudor Dimofte, Hiroyuki Fuji, Stavros Garoufalidis, Paul Gunnells and Satoshi Nawata for stimulating conversations and Satoshi for his permission to use his unpublished formulas for torus links. The author wishes to thank the organisers of the Oberwolfach workshop on low-dimensional topology and number theory for providing an opportunity to present a of this work. The author was supported by the Netherlands organization for scientific research (NWO).
2 q-Ehrhart polynomials
We present here a brief account of a -analogue of Ehrhart polynomials and prove a reciprocity theorem for them. These polynomials provide a flexible generalization of Gaussian -binomial coefficients suitable for expressing HOMFLY polynomial of complicated knots.
Our treatment relies on the standard theory of generating functions for lattice point ennumeration and Stanley reciprocity, see for example [BR07]. The present development of -Ehrhart polynomials is heavily inspired by the preprint by Chapoton [Cha13]. The special case where the polytope is generic and the linear form positive so that no polymomials in appear is due to Chapoton.
For a polytope and a linear form we consider the weighted sum
We would like to study how changes as gets scaled as . For example when then
Let be a lattice polytope and a linear form .
There exists a two variable Laurent polynomial such that for every :
Denote by the interior lattice points of the polytope then we have the following form of Ehrhart reciprocity:
By triangulating and the inclusion-exclusion principle we may reduce to the special case where is a (lattice) simplex [BR07]. For such a simplex we consider the generating series . By Theorem 3.5 on the integer point transform [BR07] we know that is a rational function in whose denominator can be factored as . Here the product runs over the set of vertices of the simplex . If the denominator had only one such factor then we could expand to obtain part 1) of the theorem. We can reduce to this case by expanding into partial fractions with respect to with coefficients in .
For the reciprocity statement of part 2) we apply Stanley’s reciprocity theorem, Theorem 4.3 in [BR07], to the generating function
In the last equality we used the fact that the left hand side is a rational function in of the special form with .
The relevant examples for our state sums are the order polytopes [Sta72]. Given a partial ordered set we consider the order polytope given by
The order polytope is a lattice polytope with vertices in the unit cube . The order polytope of a linear ordering is a simplex. The set of all linear orderings extending the partial order on the set can thus be interpreted as a triangulation of .
For example consider the order then is the triangle . We choose the linear form . The generating function for the Ehrhart polynomials is
Hence the Ehrhart polynomial is
And indeed summing the powers of in gives as expected. According to the reciprocity theorem the interior points are obtained by again as expected.
For any partial ordered set positive linear form on , the -Ehrhart polynomial does not depend on the variable .
Using the above triangulation we may reduce to the case where is a linearly ordered set, so that is a simplex. The coordinates of each vertex of the simplex are in bijection with the elements of as follows. For define as if and otherwise. It is then clear that takes different values on each of the vertices. This means that the generating series from the proof of the theorem has a denominator that is a product of distinct factors Expanding in partial fractions now shows that the only dependence of the coefficients of the series is as . ∎
As a preparation for our degree bounds for the HOMFLY polyomial we derive some elementary bounds on the degree in and for the -Ehrhart polynomials.
For any polytope with no interior vertices and linear form we have
As usual by inclusion-exclusion we may reduce to the case where is a simplex. Recall that in this case the Ehrhart series is given by
After expanding in partial fractions the highest degree term in will come from the the maximum of over the vertices of , proving part .
The proof of part is similar but we need to work a little more since the coefficients of the partial fraction expansion are rational functions in . Collect like terms in to write
where and we may assume that for . Expanding in partial fractions yields
The reader is invited to check that we have . Finally the term contributes the power to so that the replacement decreases the -degree by leaving us with . ∎
3 The symmetric MOY state sum
In this section we describe our new state sum for any link diagram in terms of MOY graphs. A MOY graph is a pair , where is an oriented trivalent graph embedded in the plane without sources or sinks together with a flow . A flow is a function such that at every vertex the sum of the values of the incoming edges equals the sum of the outgoing edges. We want to define the symmetric state sum evaluation of any MOY graph. For this we need a couple of definitions.
An elementary flow is a non-zero flow that takes values in . The set of such flows is denoted .
The orientation of induces a rotation number on each component of (i.e. for counter-clockwise for clockwise). The rotation number is the sum of the rotation numbers of the components of
The intersection number of a pair of elementary flows is defined by the formula
where is the edge at vertex that goes left respect to the orientation and goes right at .
We are now ready to define the symmetric state sum for a MOY graph. This may be regarded as a generalization of the HOMFLY polynomial to trivalent planar graphs. Including crossings in the usual way would give a generalization to any trivalent graph in the three sphere.
The symmetric evaluation of a MOY graph is defined as
Here the sum is over sequences of elementary flows and is the order polytope of interpreted as a linearly ordered set . The length of a sequence is denoted and finally the weight is .
The main result of this section is the following expression for the colored HOMFLY of any link.
Let be a link with oriented link diagram and the set of its crossings, positive and negative ones. Expand the diagram as a linear combination of MOY graphs by replacing all the crossings as shown in Figure 1, then
where the sum runs over and is the sign of the crossing .
3.1 Example: Hopf link
As a simple illustration we evaluate our state sum on the positive Hopf link colored by so we compute .
The set of crossings is and , . Expanding the two crossings as in Figure 1 we get four terms.
Since and we have only drawn one of them in Figure 2, with coefficient 2. Below each MOY graph the same figure shows the possible elementary flows and also all possible sequences of these flows that add up to the coloring. The subscript of each elementary flow is its rotation number. For example allows three different elementary flows: and the two component flow . The subscript indicates the rotation number of each flow. As shown there are only three possible sequences of elementary flows that add up to the flow of , namely and . These will be the three terms in the sum for . Since the weight and the sum of rotation numbers are we have
Here and was computed in section 2. We have
Likewise since we have
and finally so
Adding up we get the following value for the Hopf link:
in perfect agreement with the -braid formula from the introduction for .
4 The maximal degree of the colored HOMFLY polynomial
In this section we take a closer look at the terms in the state sum to prove our bounds on the maximal degree in both and . Since the colored HOMLFY polynomial not exactly a polynomial but rather a rational function of we should perhaps clarify the meaning of . For any rational function we can consider its Laurent series at infinity. It has a finite highest degree term and its exponent is what we call .
4.1 Bound on the -degree
Fix a state in the state sum. We will compute its . Since several states can cancel yields an upper bound for .
First, the term in front of the state sum contributes . Next we get from the MOY state sum. This term is computed in the following lemma:
If sums to then , where and are the numbers of positive and negative Seifert circles of the diagram.
We say two elementary flows intersect if contains edges in more than one region complementary to . Suppose our state contains a pair of intersecting flows, . We may create a new state by replacing two intersecting components by their resolution as in Seifert’s algorithm. This new state has the same number of components and the same rotation numbers. Hence we may compute the sum on a state without intersecting flows. In such states all elementary flows are parallel to the Seifert-circles of the underlying knot diagram. Around each Seifert circle there are components of elementary flows with rotation number equal to the sign of the Seifert circle. Therefore the sum equals . ∎
Finally we need to estimate . According to Lemma 2 this is the maximum of over the vertices of the order polytope . This is the sum of the flows with positive rotation number. For states where each flow has a single component this is maximal and equal to .
Adding everything up we find proving part a) of theorem 1.
As claimed in the introduction this agrees with Morton’s bound when . To properly compare the formulas we note that we do not divide by the unknot; our colored HOMFLY is unreduced. Also Morton’s variable is our and his variable is our .
4.2 Bound on the -degree
As with the -bound our strategy is to estimate by studying the states carefully.
First the framing term in front of the state sum gives . Next the choice of resolving the crossings yields . According to Lemma 3 the rotation factor gives . Next comes the weight and finally the Ehrhart polynomial. For the latter we use Lemma 2 to get a maximal contribution of since each order polytope contains the origin.
Now assume that there exists a state attaining the minimal possible weight . Combining all the contributions and the upper bound for Ehrhart this means that every crossing contributes at most
This is maximal for for negative and for positive only when . We get . In conclusion our upper bound is:
proving part of Theorem 1.
The big problem with sharpening this bound is that in the presence of a negative crossing both and can yield the maximal -degree and these terms automatically come with opposite signs so that cancellation is likely to occur. For positive diagrams we have a better chance as we will see in the next subsection.
4.3 Lower bound for positive diagrams
For positive diagrams, i.e. we derive the lower bound
announced in Theorem 1 part p). We believe this bound is actually sharp: there are no terms with higher -degree but a proof of this would require more control over the Ehrhart polynomials involved.
To prove our lower bound we show that after expanding as a Laurent series in , the coefficient of the monomial is non-zero. Notice that by Theorem 1 part a) this is actually the maximal possible degree in . Consider the coefficient of . We claim that .
To prove this, we note that the analysis of the previous subsection leading to the upper bound on the overall -degree can also be used to compute . The only difference is that we get complete control over the Ehrhart term. In order to contribute maximally to the -degree we must have . Also this joint top coefficient in and has coefficient . Setting and making all flows non-intersecting we can actually attain the rest of the upper bound in . Therefore
Our final task is to show that these terms do not cancel out. For such states the is constant and the consists of elementary flows parallel to the Seifert circles such that the absolute value of the rotation number equals the number of its components. The order is determined except for flows that are not adjacent. All such flows contribute to our top term with sign . Recall that the length of the sequence , i.e. the number of elementary flows in the state. The principle of inclusion-exclusion then determines that the total coefficient must be as required.
5 The head of the colored HOMFLY polynomial of a positive braid
For closed positive braids we can actually compute the first coefficients. We may assume the braid has at least two crossings and each generator is used at least twice.
Our strategy is to reduce the computation to braids of index . Indeed one can give a bijection between the state sum of a negative braid and the sliced braid. If is the braid then the sliced braid is
Only states where all flows have a single component contribute to the first coefficients of . Here is a closure of a positive braid.
Suppose a flow has two components and , necessarily disjoint. Since there are at least two crossings between each pair of braid lines, there must be at least four paths connecting to in the resolved braid diagram with the following property. Let be the set of diagonal edges used by . We require the four sets to be disjoint. Let be the maximal crossing parameter found on the path . On each path we can choose different sequences of elementary flows such that the -th edge of the path is contained in the -th flow. Since the flows along the path cannot be ordered linearly, the degree must drop by at least . So in total the degree will drop . The crossing parameters also make the degree drop by at least . So in conclusion the degree drops by at least . ∎
From now on all elementary flows in all states will be assumed to have one component unless stated otherwise. We will now describe an algorithm to monotonically improve the weight of a given state. A step consists of selecting two consecutive flows in the linear order of the state. After resolving the intersections between the two we set to be everything to the left and to be everything to the right. The new state is the same as the old where is replaced by in that order. If the weight is improved by at least .
The algorithm terminates on states where no flows cross and touching flows are ordered such that the left is the smaller. These are the ones that contribute to the highest -degree.
To understand the contributions to the highest terms in we need only look at those states that can be reached from a terminal state by -steps backwards. Since each step costs at least such states cannot differ greatly from the terminal states.
There is a weight preserving bijection from states contributing to the highest terms of and states contributing to the highest terms of .
There is a bijection between the crossings of and and hence also for the crossing variables and corresponding signs and powers of .
Suppose we have such a state for . Resolve the crossings of in the same way and create flows on the MOY graph for as follows. If a flow runs between braid lines we make the exact same part of the flow between braid lines of . Closing and connecting missing parts this constucts a set of flows for adding to the correct total flow on the MOY graph. The order is determined as follows: two flows between higher braid lines are always higher in the order. Flows within the same two braid lines inherit their order from their order in the state for .
This map is especially interesting in case the flows never touch more than three braid lines as is the case for the states that contribute to the first terms. In this case the map preserves the weight of the states.
The inverse map is only well defined on states of for which everything in the braid lines is higher than that in the braid lines for . Luckily this is the case for our contributing states for . The flows on the braid lines are ordered linearly and so are those between braid lines. Matching up these linear orders defines a bijection between the flows. Doing this for all yields a well defined state for . Again the weight is preserved and this map is the inverse of the other by construction. ∎
6 Proof of the MOY expansion theorem
In this section we derive our symmetric state sum from the original MOY state sum.
6.1 The original MOY state sum
Let us briefly recall the original MOY state sum for MOY graphs [MOY98]. Define an element set as follows
A MOY-state is a function such that for all edges and at each vertex the union of the values of on the incoming edges equals the union of the values of on the outgoing edges. Given a state the MOY-weight of a vertex is defined as follows. Let and be the left and right edges with respect to the orientation on then define the weight to be
Finally the rotation number of a state is defined as follows. Replace each edge by parallel copies each colored by an element of . By the above requirements we can connect edges colored by the same element of in a unique planar way to form a system of oriented simple closed curves. Each curve is colored by a single element . Define its rotation number to be if is oriented counter-clockwise and if is oriented clockwise. Now define
where the sum is over all closed curves we created. The MOY state sum is now
The MOY state sum was used in [MOY98] to express the anti-symmetric specializations of the colored HOMFLY . Here denotes the Young diagram with on column of boxes, the transpose of the -row diagram.
This implicitly determines the polynomial and hence also . Here we used the general symmetry formula for the -colored HOMFLY polynomial: where denotes the transposed partition and is the number of boxes [Zhu13].
However we cannot yet use this formula to transform the MOY state sum into a state sum for since the dependence is too implicit. In the next section we will make it explicit.
6.2 Reformulation of the MOY state sum
We now reformulate the MOY state sum to explicitly include the variable . The basic idea is to collect the MOY states with equal rotation numbers and vertex weights and sum them explicitly, giving expressions in . Those explicit sums can be written as sums over lattice points in polytopes bringing us back to the -Ehrhart polynomials. The auxilliary variable included in the Ehrhart polynomials will happily turn out cancel out for all links.
We begin by reinterpreting the MOY states as functions on the set of elementary flows. Starting with a MOY state and an elementary flow define to be the set of all such that there is no larger elementary flow such that the MOY state assigns to each of its edges. Conversely a function determines a MOY state by assigning to each edge the subset . It is easy to see that such functions are indeed in bijection with the MOY states. In what follows these sets will be identified without writing out the above bijection explicitly.
Next we collect many states with the same weight by noting that the weight of a state is already determined by the relative sizes of the function values. More precisely, a MOY state determines a sequence of elementary flows by as follows. The union is a subset of and is therefore ordered. For define to be the inverse image under of the -th element of . The point of all this is that for a MOY state inducing a sequence we have .