Approach: DAHA + BPS

Chapter 6

Exceptional Knot Homology

Here we recount the results of [EG], which extend the story of knot homologies and superpolynomials to the case of the exceptional Lie algebrae6 and its fundamental, 27-dimensional representation.

This problem poses a number of unique challenges. Section 6.1 describes how we address them by combining the differentials from [DGR, GW, GS, GGS] with the refinements of [C5, C6, AS].

In Section 6.2, we provide a proposal for and three compelling examples of the structures which define our so-calledhyperpolynomials. Finally, in Section 6.3, we describe the elements of classical singularity theory which justify our differentials; see also Appendix C.

Section 6.4 recounts the results of Section 4 of [CE], where we study the stabilization of DAHA- Jones polynomials within the Deligne-Gross exceptional series of root systems [DG].

The themes studied throughout Chapter 6 are (necessarily) of experimental nature. However, the examples we produce are convincing and merit further study.

We also face a more technical/computatational challenge. Even the ordinary (quantum group) knot invariants for e₆ have not been explicitly computed in the literature. The author R.E. has computed them for the cases considered here (unpublished) and verified their coincidence with the DAHA-Jones polynomials upon t7→q. Furthermore, no corresponding homology theory has been formally defined (other than [Web]).

We manage to overcome these obstacles by applying the technique of differentials from [DGR, Ras3] to the DAHA-Jones polynomials,q, t-counterparts of quantum knot invariants defined in [C5].

This combination is sufficiently powerful to overcome all obstacles. Here, we propose so-called hyperpolynomials fore6,27torus knot homologies, as well as produce some explicit examples.

Notation and conventions

We will use two sets of conventions: the standard DAHA conventions and conventions used in the literature on quantum group invariants (“QG”). While our calculations are performed in DAHA conventions (q, t, a), we are ultimately interested in QG conventions (q, t, u). To change DAHA→ QG, we apply the “grading change” isomorphism:

a7→ut⁻¹, q7→qt², t7→q. (6.1)

Even thoughq, tare used in both sets of conventions, whether we are referring to DAHA or QG will be contextually clear.

Furthermore, for a given knot, polynomials in QG conventions are usually associated to a Lie algebragand a representation (g-module)V. Polynomials in DAHA conventions are (equivalently) associated to a root systemRand a (dominant) weightb∈P+. The correspondence betweengand Ris via the classification of complex, semisimple Lie algebras, andbis the highest weight for V, as labeled in [B].

Now, in QG-conventions, our hyperpolynomials are Poincar´e polynomials for a (hypothetical) triply-graded vector space:

H^e6,27(K;q, t, u) := X

i,j,k

qⁱt^ju^kdimH^e_i,j,k^6,27(K). (6.2)

The usual two-variable Poincar´e polynomials are returned upon settingu= 1:

P^e6,27(K;q, t) := H^e6,27(K;q, t,1), (6.3) and we have, upon taking the graded Euler characteristic with respect tot,

P^e6,27(K;q,−1) = P^e6,27(K;q), (6.4)

i.e., these “categorify” the quantum knot invariants (2.39) fore₆,27.

This story may be translated into DAHA conventions. In light of (6.1), we may also write the hyperpolynomials in DAHA conventions:

HD^E_r,s⁶(ω₁;q, t, a) := X

i,j,k

q^j+k2 t^2i−j+k2 a^kdimH^e_i,j,k^6,27(T^r,s), (6.5)

for thesame vector space as in (6.2). Though we do not consider a DAHA analog ofP^e6,27here, we may obtain the DAHA-Jones polynomial by taking the graded Euler characteristic with respect toa:

HD^E_r,s⁶(ω₁;q, t,−1) = JDf^E_r,s⁶(ω₁;q, t). (6.6) Recall that the DAHA-Jones polynomials are t-refinements of the QG knot invariants. They are (conjecturally) related by settingt7→q:

JDf^E_r,s⁶(ω1;q, q) = P^e6,27(T^r,s;q). (6.7) Thus, we come full circle and make contact with the QG conventions at the level of polynomials.

For the convenience of the reader, our conventions and notations are summarized in the following commutative diagram:

DAHA HD ^a=−1

(6.6) //

OO

(6.1)

JDf

t7→q (6.7)

Hi,j,k (6.5)

99

(6.2)

%%QG H ^u=1_(6.3) //P ^t=−1_(6.4) //P

(6.8)

Our approach

Torus knots

Presently, our approach is confined to the torus knots and links for which the DAHA-Jones poly- nomials are defined. The reason for this limitation is algebraic from the DAHA point of view. The geometric and physical reasons were discussed in Section 3.4.

In either case, the origin of the extra grading (resp. variableu) has nothing to do with the choice of homology (Khovanov, colored HOMFLY, or other); it simply comes from a very special choice of the knot (link) and exists only for torus knots and links.

As a result, what for a generic knotK might be a doubly-graded homologyH^g,V_i,j (K) becomes a

triply-graded homologyH^g,V_i,j,k(K) for a torus knot, with an extrau-grading. Likewise, what normally would be a triply-graded (say, HOMFLY or Kauffman) homology, for a torus knotK=T^r,s becomes a quadruply-graded homologyH^g,V_i,j,k,`(T^r,s), c.f. [GGS].

Hyper-lift

We wish to elevate the two-variable DAHA-Jones polynomial JDf^E_r,s⁶(ω1;q, t), which in general has both positive and negative coefficients, to a three-variable hyperpolynomial HD_E^r,s

6(ω1;q, t, a) with only positive coefficients.

As in (6.5), this “upgraded” polynomial will be the Poincar´e polynomial of a triply-graded vector spaceH^e_i,j,k^6,27(T^r,s), accounting for its positive coefficients. As in (6.6), it is related toJDf^E_r,s⁶ by taking the graded Euler characteristic with respect to thek-grading (resp. variable a):

HD^r,s_E

6(ω₁;q, t,−1) =JDf^E_r,s⁶(ω₁;q, t). (6.9) Note that we are here constructing the polynomial, HD_E^r,s

6(ω₁), whose constituent monomials en- code the graded dimensions of the irreducible components of the vector space H_i,j,k^E6,r,s. We are not constructing this vector space itself.

Of course, there will be many polynomialsHD_E^r,s

6(ω1) that satisfy only the aforementioned prop- erties. We will define ours intelligently so that it is uniquely determined and so that like the HOMFLY-PT (“superpolynomial”) and Kauffman homologies–which respectively unify slN and soN invariants–our “hyperpolynomial” will unify the (e6,27)-invariant with invariants associated to “smaller” algebras and representations (g, V).

Differentials and specializations

This unification with other (g, V) is effected using a certain (hypothetical) spectral sequence on H^e∗^6,27 induced by deformations of the potentialW_E6,27;W_g,V, which are studied in Section 6.3.

With the additional assumption that these spectral sequences converge on their second pages, such a deformation gives rise to a differentiald_g,V such that the homology:

H_∗(H^e_∗^6,27, dg,V)∼=H^g,V_∗ . (6.10) Practically speaking, suppose that such a differentialdg,V exists (=dR,bin DAHA conventions), and that its (q, t, a)-degree is (α, β, γ). Then each monomial term in HD_E^r,s

6(ω1) will participate in exactly one of two types of direct summands in the chain complex (H^e∗^6,27, dR,b):

0−→^d qⁱt^ja^k −→^d 0, (6.11)

0−→^d qⁱt^ja^k−→^∼= q^i+αt^j+βa^k+γ −→^d 0. (6.12)

Observe that we can re-express this as a decomposition:

HD_E^r,s

6(ω1) =HDgR(b) + (1 +q^αt^βa^γ)Q(q, t, a), (6.13) whereHDgR(b) is related toJDf^R(b) by the specialization

HD_E^r,s

6(ω1;a=−q^−αγt^−βγ) =HDgR(b;a=−q^−αγt^−βγ) =JDf^R(b), (6.14) which subsumes the differential dR,b, realized by setting (1 +q^αt^βa^γ) = 0. Note that since these polynomials always have integer exponents (corresponding to integer gradings of a vector space), we will always be able to define thea-grading in such a way thatγ dividesαandβ.

To restore the a-grading toJDf^E6(ω₁), we must play this game in reverse. On the q,t level, we have a decomposition:

JDf^E6(ω1) =JDf^R(b) + (1±q^αt^β)Q(q, t). (6.15) Since many of the polynomialsJDf^R(b) are known, we can hope to use this structure to recover the a-gradings of specific generators as well as thea-degrees of thedR,b. If we can do this for sufficiently many (R, b), we will obtain enough constraints (specializations) to uniquely define the (relative) a-grading inHD^r,s_E

6(ω₁).

Uniqueness

Suppose that we have defined HD by some (possibly infinite) set of differentials/specializations S:={(R, b, α, β, γ)}, each of the form (6.13) with thesame HDg_R(b). If two hyperpolynomialsHD₁, HD₂each satisfy all of the specializations S, then evidentlyHD₁−HD₂∈I_S, where

IS:= Y

S

1 +q^αt^βa^γ

!

(6.16)

is an ideal in Γ :=Z[[q, t, a]]. Then HDcorresponds to a unique coset [HD]∈Γ/IS.

If S is infinite, then we may choose a distinguished representative of [HD], i.e., the only one with finitely many terms. This is precisely the situation when considering superpolynomials and hyperpolynomials for the classical series of Lie algebras.

When S is finite, there is also a distinguished representative. Since HD is required to have positivecoefficients, we may simply require that it is minimal in [HD] with respect to that property, i.e., it has the minimum number of terms.

Indeed, suppose HD1 6=HD2 are minimal, and write HD1−HD2 =F ·Q

S(1 +q^αt^βa^γ)∈ IS

for someF ∈R. Since theHDi both have positive coefficients, we may write F =F1−F2, where each Fi has only positive coefficients. Then clearly the monomials in Fi·Q

S(1 +q^αt^βa^γ) are all

monomials inHD_i, and since these belong toI_S, they cancel in every specialization inS. Then HD⁰_i:=HDi−Fi·Y

S

(1 +q^αt^βa^γ) (6.17)

is a new polynomial with positive coefficients and fewer terms, and which satisfies all of the special- izationsS. This contradicts the assumed minimality ofHD_i.

Restricting ourselves to these distinguished representatives, the uniqueness of our HDdepends on the uniqueness of theHDgR(b) chosen simultaneously for{(R, b)} ⊂S. As we will see below, this is manifest in all cases considered.