DAHA and plane curve singularities
We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities. This generalizes the Cherednik-Danilenko conjecture on the Betti numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of superpolynomials of torus knots and that by Gorsky-Mazin for their constant term. The paper mainly focuses on non-torus algebraic knots. A connection with the conjecture due to Oblomkov-Rasmussen-Shende is possible, but our approach is different. A motivic version of our conjecture is related to p-adic orbital A-type integrals for anisotropic centralizers.
Key words: Hecke algebra; Jones polynomial; HOMFLYPT polynomial; Khovanov-Rozansky homology; algebraic knot; Macdonald polynomial; plane curve singularity; compactified Jacobian; Puiseux expansion; orbital integral
MSC (2010): 14H50, 17B22, 17B45, 20C08, 20F36,
22E50, 22E57, 30F10, 33D52, 33D80, 57M25
- 0 Introduction
- 1 DAHA superpolynomials
- 2 Geometric superpolynomials
- 3 The family
- 4 Numerical support
- 5 Some perspectives
- A Non-affine cells
- B Dimensions for (6,9,13)
We propose a relatively simple and totally computable conjectural geometric description of uncolored DAHA superpolynomials of arbitrary algebraic knots in terms of flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities, presumably coinciding with the corresponding stable Khovanov-Rozansky polynomials. This description significantly generalizes (a) Cherednik-Danilenko’s conjecture on the Betti numbers of Jacobian factors (any unibranch singularities), (b) Gorsky’s conjectural interpretation of superpolynomials of torus knots from [Gor1] etc., and (c) that from [GM1, GM2] for their –constant term. Our conjecture is different from the ORS conjecture from [ORS, GORS] (though some connection is not impossible).
Motivation. Algebraic-geometric theory of topological invariants of algebraic links has a long history, starting with the well-known algebraic interpretation of the Alexander polynomials . This paper provides an algebro-geometric description of stable Khovanov-Rozansky polynomials via the DAHA–superpolynomials. See e.g. [KhR1, KhR2, Kh, Ras, WW]. The geometry of flagged Jacobian factors conjecturally provides the DAHA–superpolynomials from [Ch2]–[ChD2]. For instance, this explains the positivity of the latter in the uncolored case, conjectured in [ChD1].
For uncolored torus knots , this positivity results from the combinatorial construction from [Gor1, Gor2], which conjecturally provides both, stable KhR -polynomials and those via DAHA (and is closely related to rational DAHA). Our conjecture makes the positivity entirely geometric (generalizing [GM1, GM2]) for torus and arbitrary algebraic knots. We expect important implications in the theory of plane curve singularities, the theory of –adic orbital integrals and affine Springer fibers ; see Conjecture 2.5 and Section 5.
Algebraic knots. Torus-type (quasi-homogeneous) plane singularities are very special. Not much is actually known on the Jacobian factors of non-torus plane singularities; the paper [Pi] still remains the main source of examples. It was an important development when the DAHA approach from [Ch2, GN, Ch3] was extended from torus knots to arbitrary algebraic knots in [ChD1] and then to any algebraic link in [ChD2]. The Newton pairs and the theory of Puiseux expansion , the key in the topological classification of plane curve singularities, naturally emerge in the DAHA approach.
One of the key advantages of the usage of DAHA is that adding colors is relatively direct (via the Macdonald polynomials), which is well understood for any iterated torus link (including all algebraic links). This is well ahead of any other approaches (topology included) for such links. We expect that our present paper can be enhanced by adding colors via (presumably) the curves suggested in [Ma]. The case of rectangle Young diagrams is exceptional due to the conjectured positivity of the corresponding reduced DAHA superpolynomials for algebraic knots [Ch2, ChD1]. The switch from the rank-one torsion free modules in the definition of compactified Jacobians to arbitrary ranks is expected here (among other modifications), which is in progress.
The passage to arbitrary algebraic knots and links from torus knots is important because of multiple reasons. The generality is an obvious advantage, but not the only one. All algebraic links (not only torus knots) are necessary to employ the technique of the resolved conifold and similar tools used in [Ma] to prove the (colored generalization of) the OS conjecture [ObS] concerning the HOMFLYPT polynomials. Also, all algebraic links are needed for the theory of Hitchin and affine Springer fibers, since spectral curves are generally not unibranch. Topologically, the class of iterated links is closed with respect to cabling , a major operation in knot theory.
ORS Conjecture. Let us briefly comment on Conjecture 2 from [ORS]; see Section 5 for some further discussion. It relates the geometry of nested Hilbert schemes of arbitrary (germs of) plane curve singularities to the Khovanov-Rozansky unreduced stable polynomials of the corresponding links. The main component of their conjecture vs. the OS conjecture is the weight filtration . The polynomials there conjecturally coincide with uncolored ones in the -DAHA theory (upon switching to the standard parameters); see [ChD1]. They are connected with the perverse filtartion on the cohomology of the compactified Jacobians from [MY, MS] (see Proposition 4 in [ORS]).
Our approach is based on admissible flags of submodules in the normalization ring; here dim but they are not full flags and the admissibility is a very restrictive condition. The absence of (nested) Hilbert schemes is due to the reduced setting of our paper (continuing [ChD1]); there are other important deviations from [ORS]. For instance, we do not need the weight filtration and our approach is quite computable. There may be a connection with Section 9.1 from [GORS] (a reduced version of the construction of [ORS]) but this is unclear. Actually, the weight filtration appears naturally in (5.2), which follows from a modular variant (2.7) of our conjecture, but it is associated with a parameter different from that in [ORS].
Main results. The key is Conjecture 2.5; anything else is about confirmations, examples and connections. It extends Conjecture 2.4 from [ChD1] for Betti numbers of Jacobians factors for unibranch plane curve singularities (the case ). It was essentially checked in [Mel] for torus knots. The Betti numbers for torus knots are due to [LS] (see also [Pi, GM1]). We focus in this paper on non-torus knots.
The series of the plane curve singularities for Puiseux exponents for odd and is the simplest of non-torus type; the corresponding links are . Here we generalize the formulas from [Pi] for the dimensions of cells in the corresponding –presentation. The most convincing demonstrations of our Main Conjecture are the examples where such cells are not all affine. Such examples are well beyond [Pi] and are actually a new vintage in the theory of compactified Jacobians as well as our flagged generalization.
Some perspectives. An extension of the geometric approach to superpolynomials from this paper to all root systems is of obvious interest, especially due to connections with –adic orbital integrals. Paper [ChE] hints that such a uniform theory may exist, in spite of the fact that there can be no rank stabilization for the systems . Such a theory can be expected to provide refined generalizations of orbital integrals from the geometric Fundamental Lemma; local spectral curves are taken here as plane curve singularities. See the end of the paper.
The case is directly related to –adic orbital integrals of nil-elliptic type . An immediate corollary of Conjecture 2.5 is that such orbital integrals are topological invariants of the corresponding plane curve singularities. This readily follows from [LS] for torus knots, but seems beyond any existing approaches for non-torus singularities, especially in the presence of non-affine cells (see online version of this paper). This invariance, the refined orbital integrals, the connections with HOMFLYPT homology and an extension of our paper to arbitrary algebraic links, any colors and all root systems are natural challenges.
1. DAHA superpolynomials
We will provide here the main facts of DAHA theory needed for the definition of the DAHA-Jones polynomials and DAHA superpolynomials. See [Ch2, Ch3, Ch1] for details. The construction is totally uniform for any root systems and weights.
1.1. Definition of DAHA
Let be a root system of type with respect to a euclidean form on , the Weyl group generated by the reflections , the set of positive roots corresponding to fixed simple roots . The form is normalized by the condition for short roots. The weight lattice is , where are fundamental weights. The root lattice is . Replacing by , we obtain See e.g., [Bo] or [Ch1].
Setting , the vectors for form the twisted affine root system ( are identified with ). We add to the simple roots for the maximal short root . The corresponding set of positive roots is .
The set of the indices of the images of by all automorphisms of the affine Dynkin diagram will be denoted by (). Let ; for . The elements for are minuscule weights , defined by the inequalities for all . We set for the sake of uniformity.
Affine Weyl groups. Given , let
for . The affine Weyl group is the semidirect product of its subgroups and , where is identified with
Using the presentation of as , the extended Weyl group can be defined as , where the corresponding action is
It is canonically isomorphic to for . The latter group consists of id and the images of minuscule in .
The group will be naturally identified with the subgroup of of the elements of the length zero; the length is defined as follows:
One has for , where is the (unique) element of minimal length such that .
Setting for the length coincides with the length of any reduced decomposition of in terms of the simple reflections (a standard and important fact).
Let be the least natural number such that for . The double affine Hecke algebra, DAHA , depends on the parameters to be exact, it is defined over the ring of polynomials in terms of and
For , we set , , and introduce from the relation . For , let
where sht, lng are used for short and long roots. We note that the specialization corresponds to quantum groups and provides the WRT invariants in the construction below; see [Ch2].
For pairwise commutative
For instance, .
Recall that for (see above). Note that is , where is the standard involution of the nonaffine Dynkin diagram, induced by ; it is the reflection of in type . Finally, we set when the number of links between and in the affine Dynkin diagram is .
The double affine Hecke algebra is generated by the elements , pairwise commutative satisfying (1.3) and the group where the following relations are imposed:
(i) factors on each side;
(iv) if ;
Given the product
does not depend on the choice of the reduced decomposition. Moreover,
In particular, we arrive at the pairwise commutative elements
1.2. Main features
The following maps can be (uniquely) extended to automorphisms of , where must be added to the ring of constants (see [Ch1], (3.2.10)-(3.2.15)):
These automorphisms fix and their fractional powers.
The span of is the projective (due to Steinberg), which is isomorphic to the braid group . Let us list the matrices corresponding to the automorphisms above upon the natural projection onto , which is upon the specialization . The matrix will represent the map for . One has: , , .
We note that there are some simplifications with the definition of DAHA and for and in part of Theorem 1.2, but they are not significant (the theory is very much uniform for any root systems). However is obviously needed in part in this theorem.
Following [Ch1], we use the PBW Theorem to express any in the form for , (this presentation is unique). Then we substitute:
The functional , called coinvariant , acts via the projection of onto the polynomial representation , which is the –module induced from the one-dimensional character for and . Here ; see [Ch1, Ch2, Ch3].
The polynomial representation is linearly generated by and the action of there is given by the Demazure-Lusztig operators :
The elements become the multiplication operators and act via the general formula for .
Macdonald polynomials. The Macdonald polynomials for are uniquely defined as follows. For , let be a unique element such that . Given and assuming that is such that ,
Here is the constant term; is considered a Laurent series of with the coefficients expanded in terms of positive powers of . The coefficients of belong to the field . The following evaluation formula (the Macdonald Evaluation Conjecture) is important to us:
1.3. Algebraic knots
Torus knots are defined for any integers such that gcd. One has the symmetry , where we use “” for the ambient isotopy equivalence. Also for the unknot , denoted by .
Algebraic knots are associated with two sequences of (strictly) positive integers:
will be called the length of . The pairs are characteristic or Newton pairs .
We will need one more sequence:
in terms of the cabling defined below. Note that the first iteration (application of ) is for (not for the last pair!).
Cabling. The cabling of any oriented knot in (oriented) is defined as follows; see e.g. [Mo, EN] and references therein. We consider a small –dimensional torus around and put there the torus knot in the direction of , which is (up to ambient isotopy).
This procedure depends on the order of and the orientation of . We choose the latter in the standard way (compatible with almost all sources, including the Mathematica package “KnotTheory”); the parameter gives the number of turns around . This construction also depends on the framing of the cable knots; we take the natural one, associated with the parallel copy of the torus where a given cable knot sits (its parallel copy has zero linking number with this knot).
By construction, and for any knot and . See [ChD1] for further discussion of relations. The pairs are sometimes called topological; the isotopy equivalence of algebraic knots generally can be seen only at the level of –parameters (not at the level of the Newton or Puiseux pairs).
Newton-Puiseux theory. Given a sequence of Newton (characteristic) pairs the knot is the link of the germ of the singularity
which is the intersection of the corresponding plane curve with a small -dimensional sphere in around . We will always assume in this paper that this germ is unibranch.
Jacobian factors. One can associate with a unibranch the Jacobian factor . Up to a homeomorphism, it can be introduced as the canonical compactification of the generalized Jacobian of an integral rational planar curve with as its only singularity. It has a purely local definition, which we will use below. Its dimension is the –invariant of the singularity also called the arithmetic genus.
Calculating the Euler number , the topological Euler characteristic of and the corresponding Betti numbers in terms of is a challenging problem. For torus knots , one has due to [Bea]. This formula is related to the perfect modules of rational DAHA and the combinatorics of generalized Catalan numbers; see e.g. [GM1].
The Euler numbers of were calculated in [Pi] (the Main Theorem) for the following triples of Puiseux characteristic exponents :
where , and respectively. Here dim is for and for the series . Generally, equals the cardinality , where is the valuation semigroup associated with ; see [Pi] and [Za]. The Euler numbers of the Jacobian factors can be also calculated via the HOMFLYPT polynomials of the corresponding links (see below) due to [ObS, Ma].
Concerning the Betti numbers for the torus knots and the series , the odd (co)homology of vanishes. The formulas for the corresponding even Betti numbers were calculated explicitly for many values of in [Pi], where . Not much was and is known/expected beyond these two series.
1.4. DAHA-Jones theory
The construction is given directly in terms of the parameters , though it actually depends only on the corresponding topological parameters . Recall that torus knots are naturally represented by with the first column ( denotes transposition), where and we assume that gcd. Let be any pullback of to the projective .
For a polynomial in terms of fractional powers of and , the tilde-normalization will be the result of the division of by the lowest –monomial, assuming that it is well defined. We put for a monomial factor (possibly fractional) in terms of . See [ChD1] for the following theorem. We will also apply this definition to the superpolynomials, where the lowest –monomial is picked from the constant –term.
Let be a reduced irreducible root system. Recall that , where the action of in is used.
(i) Given two strictly positive sequences of length as in (1.13), we lift to and then to (acting in ) as above. For a weight , the DAHA-Jones polynomial is
It does not depend on the particular choice of the lifts and . The tilde-normalization is well defined and is a polynomial in terms of with the constant term .
(ii) Let us switch to the root system for , setting and considering as (dominant) weights for any (for ) with , where we assume that upon the restriction to .
Then given as above, there exists a DAHA–superpolynomial from satisfying the relations
then its –constant term is automatically tilde-normalized.
Topological connection. Let us briefly discuss the conjectural relation of DAHA–superpolynomials to stable Khovanov-Rozansky polynomials denoted by . See [KhR1, KhR2, Kh, Ras]. We consider only the reduced setting (actually not quite developed topologically).
The passage to the Khovanov-Rozansky polynomials for for sufficiently large is the substitution . Note the relation to the Heegaard-Floer homology for . Equivalently, this passage is in the standard topological parameters (also used in the ORS Conjecture ), which are related to the DAHA parameters as follows:
For the DAHA–superpolynomials from Theorem 1.19,
and is reduced divided by the smallest power of and then by such that with the constant term . Here means the switch from the DAHA parameters to the standard topological parameters.
Also, the polynomials are expected to coincide with the (reduced) physics superpolynomials based on the BPS states [DGR, AS, FGS, GGS] and those obtained in terms of rational DAHA [GORS, GN] for torus knots. The latter approach is developed so far only for torus knots and in the uncolored case; there is some progress for symmetric powers of the fundamental representation (see [GGS]). We will not touch the connections with rational DAHA in this paper. Concerning physics origins, let us mention that using the Macdonald polynomials at roots of unity (for ) instead of Schur functions in the usual construction of knot opertators was suggested in [AS].
Betti numbers . We are very much motivated by the DAHA approach to these numbers. Technically, we generalize the interpretation of Gorsky’s superpolynomials for torus knots at from [GM1, GM2] and the following conjecture from [ChD1]:
It implies that (the van Straten- Warmt conjecture). Relation (1.22) will be generalized below to the whole superpolynomial , which is the main result of our paper.
2. Geometric superpolynomials
2.1. Modules of semigroups
Let be the complete local ring of the unibranch germ of the plane curve singularity, embedded into the normalization ring . The conductor of is the smallest such that ; actually it is the ideal , but we will call the conductor in this paper. We set .
The corresponding semigroup is formed by the orders of the smallest powers, i.e. valuations (minimal –degrees) . The –invariant (the arithmetic genus) is then . We will call the set of gaps and denote it ; thus . Also, .
Compactified Jacobians for projective curves are generally defined as the varieties of coherent torsion free sheaves of rank one and fixed degree up to isomorphisms. The Jacobian factor , we are going to define, is a local version of the compactified Jacobian. It is (as a set) formed by all finitely generated –submodules , of (any) prescribed degree, also called –lattices.
We define –degree deg with respect to ; it is dim if . For arbitrary submodules :
This definition is a natural counterpart of the degree of a divisor at a given point (here at ) in the smooth situation. Actually we will mainly need below deg.
The valuations of the elements form a -module ; the modules for semigroups (with ) are subsets such that . Unless stated otherwise, we assume that and that it contains the element (of valuation ), such an embedding can be achieved by the division by for . Here the upper limit is sufficient in the sum due to the definition of the conductor. The notation is used for such a normalization in [Pi], we call it the standard normalization. See there for these and related definitions and facts.
For a standard , we will use the notation or for and call it the set of added gaps or simply the –set . The square brackets will be used for the list of its elements. For instance, corresponds to the (trivial) –module , and for (recall that is the last gap in ).
Due to the normalization we impose, one has:
Not all –modules can be realized as for non-torus singularities. Recall that torus knots are associated with the rings . The simplest example of a non-torus singularity is with and . Then the sets of added gaps and do not come from any modules ; following [Pi], we call them non-admissible . All other are admissible in this case (i.e. can be obtained as ).
2.2. as a projective variety