EEXCEPTIONAL KNOT HOMOLOGY
ROSS ELLIOT AND SERGEI GUKOV
Abstract.
The goal of this article is twofold. First, we find a natural home for the double affineHecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants oftorus knots and links called hyperpolynomials that address the “problem of negative coefficients”often encountered in DAHA-based approaches to homological invariants of torus knots and links.Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau-Ginzburg potentials, we also describe a rich structure of differentials that act on homological knotinvariants for exceptional groups and uniquely determine the latter for torus knots.
Contents
0. Introduction 20.1. Acknowledgements 31. Knot homologies and refined BPS states 31.1. Large N duality and BPS states 31.2. Knot invariants and topological strings 51.3. Knot homologies and refined BPS states 61.4. M-theory descriptions 72. DAHA-Jones polynomials 82.1. Affine Hecke algebras 82.2. DAHA and Macdonald polynomials 112.3. DAHA-Jones polynomials 132.4. Relation to torus knot polynomials/homologies 143. Exceptional knot homology 163.1. Approach: DAHA + BPS 163.2. E -hyperpolynomials 203.3. Computations with DAHA-Jones polynomials 213.4. Further properties 254. Adjacency tree of the corank-2 singularity Z , Date : October 11, 2018. a r X i v : . [ m a t h . QA ] M a y ROSS ELLIOT AND SERGEI GUKOV Introduction
Categorification of quantum group invariants has been a very active area of research in the pastseveral years. By now, a number of methods have been developed that allow one to “promote” apolynomial invariant P g ,V ( K ; q ) of a knot K colored by a representation V of U q ( g ) to a bi-gradedhomology theory H g ,Vi,j ( K ), whose Euler characteristic is P g ,V ( K ; q ):(0.1) P g ,V ( K ; q ) = (cid:88) i,j ( − j q i dim H g ,Vi,j ( K ) . In practice, it is often convenient to work with Poincar´e polynomials of H g ,Vi,j ( K ):(0.2) P g ,V ( K ; q, t ) = (cid:88) i,j q i t j dim H g ,Vi,j ( K ) , or, better yet, with the so-called superpolynomials P ( K ; a, q, t ) that depend on three variables andpackage homological invariants of arbitrary rank and fixed Cartan type.While formal definitions of these homological knot invariants are available for many groupsand representations [KhR, Y, Web, Wu], their calculation has been a daunting task. Besidesthe Khovanov-Rozansky homology [KhR], which corresponds to the fundamental representationof g = sl ( N ) and is reasonably computable, at present there exist only two approaches amenableto calculations for arbitrary groups and representations. One approach [DGR, GW, GS] is basedon the formal structure of knot homologies (and superpolynomials) that follows from the physicalinterpretation [GSV, G, W2] of knot homologies. Another approach [C4, C5] proposed recently isbased on DAHA (see also [AS]).Both of these approaches have advantages and disadvantages. The first approach allows one tocompute homological invariants / superpolynomials of arbitrary knots, while the second approachis limited to torus knots. On the other hand, the second approach can easily be implemented on acomputer, whereas the first approach can only be done “by hand” for simple knots with up to 10crossings or so.More importantly for the present paper, neither approach is limited to classical groups or partic-ular representations. We use this feature to tackle one of the most difficult problems in this subject:the study of homological invariants (and superpolynomials) associated with exceptional groups. Infact, for a problem like this we will need to combine the power of both methods, because each oneindividually is not sufficient for producing a polynomial with positive coefficients.The simplest “exceptional knot homology” corresponds to the minuscule 27-dimensional repre-sentation of the simply-laced Lie algebra e . The representation of the principal SL (2) on isisomorphic to the representation of the Lefschetz SL (2) on the cohomology of the 16-dimensionalflag variety G/P , with the Poincar´e polynomial, P ( t ) = 1 + t + t + t + 2 t + 2 t + 2 t + 2 t + 3 t + 2 t + 2 t + 2 t + 2 t + t + t + t + t . The strategy of our approach will be the following. First, we compute the DAHA-Jones polyno-mials of simple torus knots colored by the 27-dimensional representation of e using the approachof [C4, C5]. These will turn out to have both positive and negative coefficients. To fix this problemand to construct analogues of superpolynomials with positive coefficients, we will resort to the XCEPTIONAL KNOT HOMOLOGY 3 other method [DGR, GS, GGS] based on a rich structure of the differentials. Which differentials toexpect and how they should act is controlled by deformations of a certain singularity [GW], whichwill be yet another new result of this paper.The structure of this paper is as follows. In Section 1, we will review the physical realization ofknot homologies as spaces of BPS states in topological string theory. In Section 2, we define theDAHA-Jones polynomials and explain their relationship to torus knot polynomials and homologies.Section 3 contains our main proposal for E -hyperpolynomials, as well as three convincing ex-amples. At various intermediate stages in our calculations we shall need superpolynomials for rootsystems of Cartan types A and D . The corresponding results are summarized in Appendix A andcan be found in [GW, GS, C4]. Appendix B contains diagrams that depict our examples.Finally, in Section 4 we classify the adjacencies (infinitesimal deformations) of the singularity Z , and compute the corresponding spectra. As explained there and in [GW], deformations of thissingularity control which differentials we are to expect from ( e , ) knot homologies. The resultsof this analysis are contained in Appendix C.0.1. Acknowledgements.
Our special thanks go to Ivan Cherednik, who provided the formulasfor DAHA-Jones polynomials and participated in the development of many ideas contained herein.Without his contributions, this work would not be possible.We would also like to thank J. Adams, M. Aschbacher, D. Bar-Natan, P. Cvitanovi´c, W.A. deGraaf, A. Gabrielov, and S. Morrison for helpful discussions. The work of S.G. is funded in part bythe DOE Grant DE-SC0011632 and the Walter Burke Institute for Theoretical Physics. The workof R.E. is partially supported by a Troesh Family Graduate Fellowship 2014-15.1.
Knot homologies and refined BPS states
Large N duality and BPS states. Following [W1], recall that the Chern-Simons TQFT ona 3-manifold M with gauge group G at level k ∈ Z is described by the action functional:(1.1) S ( A ) = k π (cid:90) M tr( A ∧ dA + 23 A ∧ A ∧ A ) , where A is the ( g -valued) connection one-form of a principal G -bundle on M . The partition functionof this theory is given by the path integral,(1.2) Z ( M ) = (cid:90) A [ D A ] e iS ( A ) , over the configuration space A of principal G -connections on M . Owing to the topological natureof Chern-Simons theory, Z ( M ) is, a fortiori , a topological invariant of M .Now consider the open string theory described by the topological A-model on the cotangentbundle T ∗ M with N D-branes wrapping the Lagrangian M ⊂ T ∗ M , and coupling constant,(1.3) g s = 2 πik + N .
ROSS ELLIOT AND SERGEI GUKOV
When G = SU ( N ), it was shown in [W3] that the N expansion of the Chern-Simons free energy F ( M ) = log Z ( M ) is naturally identified with with the contribution to free energy of the degenerateinstantons in this topological string setup.Instantons there are generally described by holomorphic maps of Riemann surfaces with La-grangian boundary conditions:(1.4) (Σ , ∂ Σ) (cid:44) → ( T ∗ M, M ) . However, an easy consequence of Witten’s “vanishing theorem” is that the only such maps are thedegenerate (constant) ones. Therefore, one identifies(1.5) Z CS ( M ) = Z openstring ( T ∗ M ) , the partition functions for Chern-Simons gauge theory on M and the open topological string theoryon T ∗ M .In the special case of M = S , it was conjectured [GV] that at large N , this open string setupundergoes a geometric transition which produces a (physically equivalent) closed string theory. This conifold transition shrinks the 3-cycle of the deformed conifold T ∗ S to a point and resolves theresulting conical singularity with a small blow-up. The resulting space X is the resolved conifold ,i.e. the total space of the O ( − ⊕ O ( −
1) bundle over C P .Observe that the conifold transition eliminates the N branes wrapping S , producing a closedstring theory on X . In the worldsheet description of this theory, based on the genus g topologicalsigma model coupled to 2-dimensional gravity, the free energy is(1.6) F g ( t ) = (cid:88) Q ∈ H ( X ) N g,Q e − tQ , where the parameter t is the K¨ahler modulus for the Calabi-Yau space X :(1.7) t = 2 πiNk + N = vol( C P ) , and N g,Q is the Gromov-Witten invariant “counting” holomorphic maps of genus g representingthe integral 2-homology class Q .The numbers N g,Q are rational, in general. However, as shown in [GV], this model also admits atarget space description in which the all-genus free energy is naturally described in terms of integerinvariants n sQ ∈ Z :(1.8) F ( g s , t ) = ∞ (cid:88) g =0 g g − s F g ( t ) = (cid:88) Q ∈ H ( X ) ,s ≥ n sQ (cid:88) m ≥ m (cid:16) mg s (cid:17) s − e − mtQ , which encode degeneracies of the so-called BPS states .In a general supersymmetric quantum theory, a BPS state is one whose mass is equal to thecentral charge of the supersymmetry algebra. In the case at hand, a state is a D2-brane wrapping C P , and the BPS condition means that it is supported on a a calibrated 2-submanifold of theCalabi-Yau X (i.e. on a holomorphic curve in X ). XCEPTIONAL KNOT HOMOLOGY 5
Thus, a minimally embedded surface representing Q ∈ H ( X ; Z ) gives rise to a component ofthe Hilbert space H BP S , i.e. a projective unitary representation of the spatial rotation group,(1.9) SO (4) ∼ SU (2) L × SU (2) R , of R obtained upon compactification from M-theory. This representation can be specified by twohalf-integer charges j L , j R ∈ Z ≥ , which are the weights of the respective SU (2) representations.One might be tempted to introduce integers n ( j L ,j R ) Q counting these states. However as onedeforms the theory, BPS states can combine into non-BPS states, so these numbers are not invariant.On the other hand, the index,(1.10) n j L Q := (cid:88) j R ( − j R (2 j R + 1) n ( j L ,j R ) Q , is well-defined on the moduli of X . The integers n sQ are then related by a change of basis for therepresentation ring of SU (2).1.2. Knot invariants and topological strings.
For a knot K ⊂ M and a representation V of g , one can consider the holonomy of A along K traced in V , yielding the gauge-invariant Wilsonloop operator:(1.11) W KV ( A ) = tr V (cid:20) P exp (cid:73) K A (cid:21) . Expanding the correlation function of a Wilson loop in q := e πik + h ∨ produces an integer Laurentpolynomial:(1.12) P g ,V ( M, K ; q ) := (cid:10) W KV (cid:11) M = 1 Z ( M ) (cid:90) A [ D A ] e iS ( A ) W KV ( A ) , which is naturally an isotopy invariant of K ⊂ M . In what follows, we will exclusively consider K ⊂ S and suppress M . Then P g ,V ( K ; q ) are the quantum knot invariants discussed in theintroduction and whose categorifications (0.1) we will discuss below.As explained in [OV], Wilson loops can be incorporated in the open string on the deformedconifold by introducing L K ⊂ T ∗ S , the conormal bundle to K ⊂ S . In particular, L K is aLagrangian submanifold of T ∗ S , which is topologically S × R and with L K ∩ S = K . Wrapping M “probe” branes on L K produces a theory with three kinds of strings:(1) both ends on S (cid:59) SU ( N ) Chern-Simons theory on S ,(2) both ends on L K (cid:59) SU ( M ) Chern-Simons theory on L K ,(3) one end on each S and L K (cid:59) complex SU ( N ) ⊗ SU ( M ) scalar field on K .Let U , V be the holonomies around K of gauge fields A , A (cid:48) in (1),(2) respectively. Then the lastkind of string (3) contributes to the overall action by(1.13) S ( U, V ) := ∞ (cid:88) n =1 n tr U n tr V − n = log (cid:34)(cid:88) R tr R U tr R V − (cid:35) . In turn, the effective action for the theory on S is(1.14) S ( A ; K ) := S CS ( A ; S ) + S ( U, V ) , ROSS ELLIOT AND SERGEI GUKOV and integrating A out of the overall theory involves evaluating(1.15) (cid:104) S ( U, V ) (cid:105) S = 1 Z ( S ) (cid:90) A [ D A ] e iS ( A ; K ) = (cid:88) λ (cid:104) W Kλ (cid:105) (tr λ V − )for fixed V , which produces a generating functional for all Wilson loops associated to K ⊂ S (i.e.for all Young diagrams λ ).If one follows the Lagrangian L K ⊂ T ∗ S through the conifold transition, the result is anotherLagrangian L (cid:48) K ⊂ X , where the M branes will still reside. In the resulting open string theory,the worldsheet perspective again “counts”, in an appropriate sense, holomorphic maps of Riemannsurfaces with Lagrangian boundary conditions:(1.16) (Σ , ∂ Σ) (cid:44) → ( X, L (cid:48) K ) , described by the open Gromov-Witten theory.From the target space perspective, states correspond to configurations in which D2-branes wraprelative cycles Q ∈ H ( X, L (cid:48) K ; Z ) and end on D4-branes which wrap L (cid:48) K . BPS states are thenminimally-embedded surfaces Σ ⊂ X with boundaries ∂ Σ ⊂ L (cid:48) K .In [OV], the authors also showed that the generating functional for Wilson loops has an inter-pretation in terms of BPS degeneracies:(1.17) (cid:104) S ( U, V ) (cid:105) S = i (cid:88) R,Q,s N R,Q,s (cid:88) m ≥ e m ( − tQ + isg s ) m sin (cid:0) mg s (cid:1) tr R V m , where N R,Q,s ∈ Z are certain modifications of n sQ . One can then express the quantum invariant P sl N ,R ( K ; q ) directly in these terms. For example, if R = (cid:3) we have(1.18) P N ( K ; q ) = 1 q − q − (cid:88) Q,s N (cid:3) ,Q,s q NQ + s , directly relating quantum knot invariants to the enumerative geometry of X .1.3. Knot homologies and refined BPS states.
In light of the mathematical development ofhomology theories categorifying quantum knot invariants, one might ask whether they also admitphysical descriptions in the contexts outlined above. This program was initiated in [GSV], wherethe authors refined the BPS degeneracies:(1.19) N (cid:3) ,Q,s ( K ) = (cid:88) r ( − r D Q,s,r ( K ) , introducing non-negative integers D Q,s,r ∈ Z ≥ , which also reflect the charge r of U (1) R ∈ SU (2) R .Given that the Calabi-Yau X is rigid, these numbers are invariant under complex structure defor-mations. Furthermore, as we mentioned earlier in (1.7), the K¨ahler modulus of X is related to the rank of the underlyingroot system via q N = e t = exp(vol( C P )), so that changes in the BPS spectrum as one varies the K¨ahler parameter t (a.k.a. the ‘stability parameter’) reflect changes of homological knot invariants at different values of N . See [GS]for details. XCEPTIONAL KNOT HOMOLOGY 7
This led to a conjecture relating the knot homology which categorifies P N ( K ; q ) to refined BPSdegeneracies:(1.20) ( q − q − ) KhR N ( K ; q, t ) = (cid:88) Q,s,r D Q,s,r ( K ) q NQ + s t r , for sufficiently large N , where KhR N ( K ; q, t ) is the Poincar´e polynomial for the Khovanov-Rozanskyhomology.More generally, one might view the charges Q, s, r as gradings on the Hilbert space H BPS ( K )and conjecture an isomorphism of graded vector spaces:(1.21) (cid:77) i,j H i,j ( K ) = H knot ( K ) ∼ = H BPS ( K ) = (cid:77) Q,s,r H Q,s,r ( K ) , with dim H Q,s,r ( K ) = D Q,s,r ( K ). This new perspective has revealed hidden structures of knothomologies that are manifest in the context of BPS states. In particular, H knot ( K ) should: • stabilize in dimension for sufficiently large N • be triply-graded, with the additional grading (corresponding to Q ) encoding N -dependenceof the homology theory • include the structure of differentials (c.f. Section 3.1) corresponding to wall-crossing behav-ior of H BPS ( K )and, in fact, all of these structures were realized in [DGR], where the authors proposed a triply-graded homology theory categorifying the HOMFLY polynomial. Furthermore, they were able toconstruct explicit Poincar´e polynomials for this homology theory (“superpolynomials”) based ona rigid structure of differentials, which was later formalized in [R]. Similar constructions for otherchoices of ( g , V ) were proposed in [GW, GS, GGS].1.4. M-theory descriptions.
M-theory on an eleven-dimensional space-time incorporates thevarious (equivalent) versions of string/gauge theory and the dualities between them. The individualtheories can then be recovered by integrating out the dependence of M-theory on some portion ofthe background geometry.Naturally, this framework can offer several equivalent but nontrivially different points of view onthe same object. In the case of knot homologies, we are looking for new descriptions of(1.22) H knot ( K ) ∼ = H BPS ( K ) , so we promote the topological string setups described above.In particular, the five-brane configuration relevant to the physical description of the ( sl N , λ ) knothomologies on the deformed conifold is:space-time : R × T ∗ S × M N M5-branes : R × S × D (1.23) | λ | M5-branes : R × L K × D ROSS ELLIOT AND SERGEI GUKOV and the equivalent (large- N dual) configuration on the resolved conifold is:space-time : R × X × M (1.24) | λ | M5-branes : R × L (cid:48) K × D where states correspond to configurations in which M2-branes wrap relative cycles Q ∈ H ( X, L (cid:48) K ; Z ),fill R , and end on the M5-branes.The precise form of the 4-manifold M and the surface D ⊂ M is not important (in mostapplications D ∼ = R and M ∼ = R ), as long as they enjoy a U (1) F × U (1) P symmetry action,corresponding to the charges that comprise the ( s, r )-gradings. The first (resp. second) factor isa rotation symmetry of the normal (resp. tangent) bundle of D ⊂ M . Following [W2], let usdenote the corresponding quantum numbers by F and P . These quantum numbers were denoted,respectively, by 2 S and 2( S − S ) in [AS] and by 2 j and n in [GS].This description of H BPS ( K ) in the M-theory framework led to a number of developments whichshed light on various aspects of knot homologies and yield powerful computational techniques.Some examples include: • [W2] formulates the relevant space of BPS states within (1.24) • [AS] refines torus knot invariants directly within Chern-Simons theory based on its rela-tionship with (1.24) discovered in [W3] • [DGH] takes the perspective of M on which the BPS invariants are expressed via equivariantinstanton counting 2. DAHA-Jones polynomials
Given the ( r, s )-torus knot, a root system R , and a weight b , the corresponding DAHA-Jonespolynomial is defined by the simple formula:
J D
Rr,s ( b ; q, t ) := { (cid:98) γ r,s ( P b ) /P b ( q − ρ k ) } ev We will briefly explain the meaning of this expression and then describe its properties and relationsto torus knot polynomials and homologies.Good general references for the material in this section are [C3, Ha, Hu, Ki, M1, M4] as well asthe original papers [C1, C4, C5, M2, M3]. Our conventions for root systems will be from [B].2.1.
Affine Hecke algebras.
Hecke algebras.
Let R be a (crystallographic) root system of rank n with respect to theEuclidean inner product ( − , − ) on R n , and let ∆ = { α , . . . , α n } be any set of simple roots. The Weyl group W for R is generated by the simple reflections:(2.1) s i : β (cid:55)→ β − β, α i )( α i , α i ) α i for 1 ≤ i ≤ n, β ∈ R, subject to the Coxeter relations ( s i s j ) m ij = 1. The numbers m ij are 2,3,4,6 when the correspondingnodes in the Dynkin diagram for R are joined by 0,1,2,3 edges, respectively. XCEPTIONAL KNOT HOMOLOGY 9
The (nonaffine)
Hecke algebra H for R is generated over C ( t , . . . , t n ) by elements { T , . . . , T n } ,subject to relations:(2.2) ( T i − t i )( T i + t − i ) for 1 ≤ i ≤ n, (2.3) T i T j T i . . . = T j T i T j . . . with m ij terms on each side , where the number of distinct t i is equal to the number the orbits of W acting on R , so at most 2in the nonaffine case. That is, we normalize the form by ( α, α ) = 2 for short roots α ∈ R and set ν β := ( β,β )2 for β ∈ R . Then t i := t ν αi for each simple root α i ∈ ∆.2.1.2. Twisted affine root systems.
Before defining an affine root system, we recall the identification R n +1 ∼ = Aff( R n ). That is, we interpret a vector [ (cid:126)u, c ] ∈ R n × R as an affine linear function on R n :(2.4) [ (cid:126)u, c ] : (cid:126)v (cid:55)→ ( (cid:126)u, (cid:126)v ) − c, whose zero set [ (cid:126)u, c ] − (0) is an affine hyperplane in R n , H [ (cid:126)u,c ] := { (cid:126)v ∈ R n : ( (cid:126)u, (cid:126)v ) = c } . Observethat H [ (cid:126)u,c ] = H [ (cid:126)u, + c (cid:126)u ∨ , where (cid:126)u ∨ := (cid:126)uν (cid:126)u .The reflection of R n through H [ (cid:126)u,c ] is(2.5) s [ (cid:126)u,c ] : (cid:126)v (cid:55)→ (cid:126)v − [( (cid:126)u, (cid:126)v ) − c ] (cid:126)u ∨ , which fixes H [ (cid:126)u,c ] and maps 0 to c(cid:126)u ∨ . We can extend the domain of affine reflections to act onAff( R n ) ∼ = R n × R by(2.6) s [ (cid:126)u,c ] ([ (cid:126)v, k ]) := [ (cid:126)v, k ] ◦ s [ (cid:126)u,c ] = [ (cid:126)v, k ] − ( (cid:126)v, (cid:126)u ∨ )[ (cid:126)u, c ] . Alternatively, we could describe s [ (cid:126)u,c ] as a reflection in H [ (cid:126)u, with a subsequent translation by c(cid:126)u ∨ ,where “translations” are(2.7) s [ ± (cid:126)u,c ] s [ (cid:126)u, = s [ (cid:126)u, s [ ∓ (cid:126)u,c ] : (cid:126)v (cid:55)→ (cid:126)v ± c(cid:126)u ∨ , [ (cid:126)v, k ] (cid:55)→ [ (cid:126)v, k ± ( (cid:126)v, (cid:126)u ∨ ) c ] , and we will often confuse c(cid:126)u ∨ ∈ R n with this action below.Define the (twisted) affine root system R ⊂ (cid:101) R by:(2.8) (cid:101) R = { [ α, kν α ] : α ∈ R, k ∈ Z } , with R = { [ α, } . The simple roots for (cid:101) R are (cid:101) ∆ := { α = [ − ϑ, } ∪ ∆, where ϑ ∈ R is the highest short root with respect to ∆.2.1.3. Affine Weyl groups.
The affine Weyl group (cid:102) W is generated by s i := s α i , 0 ≤ i ≤ n subject torelations s i = 1 and(2.9) s i s j s i . . . = s j s i s j . . . with m ij terms on each side , where m ij correspond, as above, to the affine Dynkin diagram.We saw that s [ α,kν α ] admits a description as a reflection s α ∈ W composed with a translation by kν α α ∨ = kα ∈ Q , where Q is the root lattice for R , i.e., the Z -span of ∆. Therefore, one easily concludes that(2.10) (cid:102) W = W (cid:110) Q, where Q acts by “translations” as described above.If we enlarge the group Q to include translations by the weight lattice,(2.11) Q ⊂ P := n (cid:77) i =1 Z ω i , where { ω i } are fundamental weights, we obtain the extended affine Weyl group ,(2.12) (cid:99) W := W (cid:110) P = (cid:102) W (cid:110) Π , where Π := P/Q in the semidirect product decomposition relative to (cid:102) W .To describe the subgroup Π (cid:47) (cid:99) W more explicitly, we can introduce a length function l on (cid:99) W :(2.13) l ( (cid:98) w ) := (cid:12)(cid:12)(cid:12) (cid:101) R + ∩ (cid:98) w ( − (cid:101) R + ) (cid:12)(cid:12)(cid:12) , where (cid:98) R + is the set of positive roots with respect to (cid:101) ∆. Then Π = { (cid:98) w ∈ (cid:99) W : l ( (cid:98) w ) = 0 } .Geometrically, these these are the elements of (cid:99) W which permute (cid:101) ∆, and we can label an element π r ∈ Π by its action π r ( α ) = α r .Alternatively, define the set of indices of minuscule weights:(2.14) O (cid:48) := { r : 0 ≤ ( ω r , α ∨ ) ≤
1, , for all α ∈ R + } ⊂ { , . . . , n } . Then O = { } ∪ O (cid:48) is a system of representatives for P/Q in the sense that every b ∈ P can bewritten uniquely as b = ω r + α for some r ∈ O , α ∈ Q , where ω = 0. For r ∈ O let u r ∈ W be theshortest element such that u r ( ω r ) ∈ − P + . We can define(2.15) Π = { π r : ω r = π r u r , r ∈ O } , and observe that π = id.The affine Weyl group (cid:102) W (or, to be more precise, its group algebra) has a simple physicalinterpretation [GW2] as the algebra of line operators in four-dimensional gauge theory on M ∼ = S × R in the presence of ramification along D ∼ = S × R . (In physics, ramification is often calleda surface operator .)2.1.4. Affine Hecke algebras.
The affine Hecke algebra H for R ⊂ (cid:101) R is generated over C ( t , t , . . . , t n ).It admits two equivalent descriptions, each emphasizing one of the two equivalent descriptions ofthe extended affine Weyl group (cid:99) W : • For (cid:99) W = (cid:102) W (cid:110) Π, H is generated by elements { T , T , . . . , T n } and π r ∈ Π, subject torelations:(1) ( T i − t i )( T i + t − i ) for 0 ≤ i ≤ n ,(2) T i T j T i . . . = T j T i T j . . . with m ij terms on each side,(3) π r T i π − r = T j if π r ( α i ) = α j . • For (cid:99) W = W (cid:110) P , H is generated by { T , . . . , T n } and { Y b : b ∈ P } , subject to relations:(1) ( T i − t i )( T i + t − i ) for 1 ≤ i ≤ n , XCEPTIONAL KNOT HOMOLOGY 11 (2) T i T j T i . . . = T j T i T j . . . with m ij terms on each side,(3) Y b + c = Y b Y c for b, c ∈ P ,(4) T i Y b = Y b Y − α i T − i if ( b, α ∨ i ) = 1 for 0 ≤ i ≤ n ,(5) T i Y b = Y b T i if ( b, α ∨ i ) = 0 for 0 ≤ i ≤ n .To translate from the first to the second description, one can define pairwise-commuting elements:(2.16) Y b := n (cid:89) i =1 Y l i i for b = n (cid:88) i =1 l i ω i ∈ P, where Y i := T ω i for ω i ∈ (cid:99) W . That is, if l = l ( (cid:101) w ) so that (cid:101) w = s i l · · · s i ∈ (cid:102) W is a reduceddecomposition, then T π r (cid:101) w := π r T i l · · · T i . For example, Y ϑ = T T s ϑ .Much like the affine Weyl group, the affine Hecke algebra H can also be interpreted as the algebraof line operators in 4d gauge theory on M with a ramification (surface operator) along D ⊂ M .The only difference is that now one has to introduce the so-called Ω-background in the normalbundle of D . (See [G2] for a review.)2.2. DAHA and Macdonald polynomials.
Double affine Hecke algebras.
Let m be the least natural number satisfying ( P, P ) ⊂ m Z .Suppose that (cid:101) b = [ b, j ] with b = n (cid:88) i =1 l i ω i ∈ P and j ∈ m Z . Then for { X , . . . , X n : [ X i , X j ] = 0 } wedefine elements:(2.17) X (cid:101) b := n (cid:89) i =1 X l i i q j , and an action of (cid:98) w ∈ (cid:99) W by (cid:98) w ( X (cid:101) b ) := X (cid:98) w ( (cid:101) b ) . Observe that X := X α = qX − ϑ .The double affine Hecke algebra (“DAHA”) HH for R ⊂ (cid:101) R is generated over Z q,t := Z [ q m , t ν ] byelements { T i , X b , π r : 0 ≤ i ≤ n , b ∈ P , r ∈ O } subject to relations:(1) ( T i − t i )( T i + t − i ) for 0 ≤ i ≤ n ,(2) T i T j T i . . . = T j T i T j . . . with m ij terms on each side,(3) π r T i π − r = T j if π r ( α i ) = α j ,(4) T i X b = X b X − α i T − i if ( b, α ∨ i ) = 1 for 0 ≤ i ≤ n ,(5) T i X b = X b T i if ( b, α ∨ i ) = 0 for 0 ≤ i ≤ n ,(6) π r X b π − r = X π r ( b ) = X u − r ( b ) q ( ω ι ( r ) ,b ) for r ∈ O (cid:48) ,where in (6) we have used the involution ι : O (cid:48) → O (cid:48) defined by π − r = π ι ( i ) .Observe that HH contains two subalgebras isomorphic to the affine Hecke algebra H for R ⊂ (cid:101) R : H := (cid:104) π r , T , . . . , T n (cid:105) ⊂ HH , (2.18) H := (cid:104) T , . . . , T n , X b (cid:105) ⊂ HH . (2.19)One can make H look more like H by defining pairwise-commuting elements Y b as in (2.16). Thenwe have that(2.20) H = (cid:104) T , . . . , T n , Y b (cid:105) . In fact, HH is also generated by elements { X a , T w , Y b : a, b ∈ P , w ∈ W } . While relations betweenthese generators are more complicated, this presentation has some nice properties that will be usefulin our definitions of Macdonald and DAHA-Jones polynomials below. In particular, we have thePBW theorem for DAHA. Theorem 2.1. (PBW Theorem) Any h ∈ HH can be written uniquely in the form (2.21) h = (cid:88) a,w,b c a,w,b X a T w Y b , for c a,w,b ∈ Z q,t . The similar statement holds for each ordering of { X a , T w , Y b } . Much like the affine Weyl group and the affine Hecke algebra, the double affine Hecke algebra HH can be interpreted as the algebra of line operators in the presence of ramification (surface operator)[G2].2.2.2. Polynomial representation.
To define the Macdonald polynomials using DAHA, we need the polynomial representation (2.22) (cid:37) : HH → V , where V := End( Z q,t [ X ]). In generators { X b , π r , T i } its action is given by(2.23) (cid:37) : X b · g = X b gπ r · g = π r gπ − r , where, e.g., π r · X b = X π r ( b ) T i · g = (cid:98) T i g , for g ∈ Z q,t [ X ]. The action of T i is by the Demazure-Lusztig operators:(2.24) (cid:98) T i := t i s i + ( t i − t − i ) s i − X α i − , where, again, s i X b = X s i ( b ) . Observe that if g ∈ Z q,t [ X ] W is any symmetric polynomial, then (cid:98) T i g = t i g . Remarkably, (cid:37) is a faithful representation.2.2.3. Symmetric Macdonald polynomials.
The symmetric Macdonald polynomials P b ∈ Z q,t [ X ] for b ∈ P + were introduced in [M2, M3]. They form a basis for the symmetric ( W -invariant) polynomi-als Z q,t [ X ] W . DAHA provides a uniform construction of P b for any root system as the simultaneouseigenfunctions for a commuting family of W -invariant operators L f for f ∈ Z q,t [ Y ] W = Z ( H ), see[C1].Now for f ∈ Z q,t [ Y ] W ⊂ HH , we can use the polynomial representation to write an operator L f := (cid:37) ( f ) on Z q,t [ X ]. The symmetric Macdonald polynomials are uniquely defined by(2.25) L f ( P b ) = f ( q ρ k + b ) P b , as simultaneous eigenfunctions of the pairwise-commuting W -invariant operators L f for all f ∈ Z q,t [ Y ] W . In fact, P b ∈ Q ( q, t ν )[ X ] W . XCEPTIONAL KNOT HOMOLOGY 13
In expressing P b as an eigenfunction, we used the notation(2.26) ρ k := 12 (cid:88) α ∈ R + k α α = k sht ρ sht + k lng ρ lng , where, e.g., ρ sht(lng) := 12 (cid:88) α short(long) k α α, for the Weyl vector weighted by a function k α = k ν α which is invariant on W -orbits. We also usethe notation X b ( q a ) := q ( b,a ) , and in particular, X b ( q ρ k ) = q ( b,ρ k ) = t ( b,ρ sht )sht t ( b,ρ lng )lng . Following [C2],we have the duality and evaluation formulas: P b ( q c − ρ k ) P c ( q − ρ k ) = P c ( q b − ρ k ) P b ( q − ρ k ) for b, c ∈ P − , (2.27) P b ( q − ρ k ) = q − ( ρ k ,b ) (cid:89) α ∈ R + ( α ∨ ,b ) − (cid:89) j =0 (cid:32) − q jα t α X α ( q ρ k )1 − q jα X α ( q ρ k ) (cid:33) . (2.28)The corresponding spherical polynomial is P ◦ b := P b /P b ( q − ρ k ).2.3. DAHA-Jones polynomials.
Here we provide an efficient definition of the DAHA-Jonespolynomials, which were originally defined in [C4, C5] for torus knots and extended to iteratedtorus knots in [CD]. We also state their main (algebraic) properties, which were conjectured in[C4] and mostly proved in [C5, GN].2.3.1.
P SL ∧ ( Z ) -action. Define a central idempotent:(2.29) e := 1 | W | (cid:88) w ∈ W w, in the group algebra of W . Then the spherical DAHA is SH := e HH e ⊂ HH . In particular, P ◦ b ∈ SH .Further, define the projective P SL ( Z ) by(2.30) P SL ∧ ( Z ) := (cid:104) τ ± : τ + τ − − τ + = τ − − τ + τ − − (cid:105) , as a group whose action HH is represented by:(2.31) τ + = (cid:32) (cid:33) , τ − = (cid:32) (cid:33) , where (cid:32) a bc d (cid:33) : X λ (cid:55)→ X aλ Y cλ T i (cid:55)→ T i Y λ (cid:55)→ X bλ Y dλ , for λ ∈ P , i > HH , which restricts to an action on SH ⊂ HH .2.3.2.
Evaluation coinvariant.
We define a functional {·} ev : HH → Z q,t called the evaluation coin-variant which first writes h ∈ HH ,(2.32) h = (cid:88) a,w,b c a,w,b X a T w Y b , in the unique form guaranteed by the PBW Theorem 2.1 and then substitutes(2.33) X a (cid:55)→ q − ( ρ k ,a ) , T i (cid:55)→ t i , Y b (cid:55)→ q ( ρ k ,b ) . This process factors through the polynomial representation, which allows one to avoid makingdirect use of the PBW theorem (which can be rather complicated to implement). In other words, {·} ev is equivalent to projection onto the polynomial representation followed by the substitution(2.33). See [CM].2.3.3. Main definition.
Corresponding to the ( r, s )-torus knot, choose an element (cid:98) γ r,s ∈ P SL ∧ ( Z )which is any word in τ ± that can be represented by(2.34) γ r,s = (cid:32) r ∗ s ∗ (cid:33) , where the ∗ entries do not matter, since (cid:98) γ r,s will act on a polynomial in X i , see (2.31). For anyroot system R and dominant weight b ∈ P + , let JD Rr,s ( b ; q, t ) := { (cid:98) γ r,s ( P b ) /P b ( q − ρ k ) } ev , (2.35) (cid:102) JD Rr,s ( b ; q, t ) := q • t • JD Rr,s ( b ; q, t ) , (2.36)where q • t • is the lowest q, t -monomial in JD Rr,s ( b ; q, t ), if it is well-defined. Then (cid:102) JD Rr,s ( b ; q, t ) ∈ Z [ q, t ]is the (reduced, tilde-normalized) DAHA-Jones polynomial .2.3.4.
Properties of DAHA-Jones polynomials.
Here we recall some important properties of DAHA-Jones polynomials, which were conjectured in [C4] and proved in Theorem 1.2 of [C5]. First, weremark that the tilde-normalized DAHA-Jones polynomials are, in fact, polynomials:(2.37) (cid:102) JD Rr,s ( b ; q, t ) ∈ Z [ q, t ] . Then, in anticipation of a connection to quantum knot invariants, we expect that DAHA-Jonespolynomials should satisfy the usual topological properties with respect to the torus knot T r,s :(1) (well-defined) (cid:102) JD Rr,s ( b ; q, t ) does not depend on the choice of (cid:98) γ r,s ∈ P SL ∧ ( Z ),(2) (unknot) (cid:102) JD Rr, ( b ; q, t ) = 1,(3) ( r, s -symmetry) (cid:102) JD Rr,s ( b ; q, t ) = (cid:102) JD Rs,r ( b ; q, t ),(4) (orientation) (cid:102) JD Rr,s ( b ; q, t ) = (cid:102) JD R − r, − s ( b ; q, t ),(5) (mirror image) JD Rr, − s ( b ; q, t ) = JD Rr,s ( b ; q − , t − ).Finally, the following evaluation is a property of the refinement which reflects “exponential growth”in the number of terms in (cid:102) JD Rr,s ( b ; q, t ) with respect to | b | :(2.38) (cid:102) JD Rr,s ( n (cid:88) i =1 b i ω i ; q = 1 , t ) = n (cid:89) i =1 (cid:102) JD Rr,s ( ω i ; q = 1 , t ) b i . It is related to the fact that P b + c = P b P c upon q →
1. We do not discuss the color exchange , whichis also part of Theorem 1.2 and corresponds to generalized level-rank duality.2.4.
Relation to torus knot polynomials/homologies.
Quantum groups.
In [C4] it was demonstrated for A , announced for A n , and conjecturedfor general root systems that, upon t (cid:55)→ q , the DAHA-jones polynomials coincide (up to an overallfactor) with the corresponding (normalized/reduced) quantum invariants of torus knots:(2.39) q • (cid:102) JD Rr,s ( b ; q, q ) conj. == P g ,V b ( T r,s ; q ) . XCEPTIONAL KNOT HOMOLOGY 15
Here g is the Lie algebra corresponding to the root system R , and V b is the representation of g withhighest weight b ∈ P + ( R ).In the author’s (R.E.) Ph.D. thesis, this connection has been established for R of types A and D , as well as for the examples of ( E , ω ) used in this paper.2.4.2. DAHA-superpolynomials.
Here we restrict to type- A root systems and present the “threesuper-conjectures” from Section 2.2 of [C4], which are now theorems due to [C5, GN]. Theorem 2.2.
For any n ≥ m − , we may naturally interpret λ ∈ P + ( A m ) as a weight for A n . (1) (Stabilization) There exists a unique polynomial HD r,s ( λ ; q, t, a ) ∈ Z [ q, t ± , a ] , defined bythe (infinitely many) specializations (2.40) HD r,s ( λ ; q, t, a (cid:55)→ − t n +1 ) = (cid:102) JD A n r,s ( λ ; q, t ) , for n ≥ m − . We will call HD r,s ( λ ; q, t, a ) the DAHA-superpolynomial. (2) (Duality) Let q A t B be the greatest q, t -monomial in HD r,s ( λ ; q, t, a ) whose a -degree is .Then (2.41) HD r,s ( λ tr ; q, t, a ) = t A q B HD r,s ( λ ; t − , q − , a ) , where λ tr indicates the transposed Young diagram for λ . (3) (Evaluation) It immediately follows from (2.38) that (2.42) HD r,s (cid:16) m (cid:88) i =1 λ i ω i ; 1 , t, a (cid:17) = m (cid:89) i =1 (cid:16) HD r,s ( ω i ; 1 , t, a ) (cid:17) λ i . When combined with the duality, this implies (2.43) HD r,s (cid:16) m (cid:88) i =1 λ i ω i ; q, , a (cid:17) = m (cid:89) i =1 (cid:16) HD r,s ( ω i ; q, , a ) (cid:17) λ i . Currently, the latter has no direct interpretation in terms of Macdonald polynomials or theDAHA-Jones construction.
We can generally make contact with the conventions used in the literature on superpolynomials,e.g., [DGR], by a transformation DAHA (cid:55)→
DGR:(2.44) t (cid:55)→ q , q (cid:55)→ q t , a (cid:55)→ a t. Then we have the following conjecture, which extends the one from [AS].
Conjecture 2.3.
For a rectangular
Young diagram i × j , i.e., a weight jω i ∈ P + , the coefficients of HD r,s ( jω i ; q, t, a ) are positive integers. In this case, upon the transformation (2.44), one recoversthe superpolynomials from [DGR, GS, GGS].In light of Conjecture 2.3, one can attribute the duality to the “mirror symmetry” and theevaluation to the “refined exponential growth” of [GS, GGS]. Furthermore, in Lemma 2.8 of [GN]the authors demonstrate that the DAHA-Jones polynomials are proper (formal) generalizations–toany root system and weight–of the refined torus knot invariants of [AS]. Exceptional knot homology
Approach: DAHA + BPS.
In [DGR] the authors introduce the superpolynomial for knothomologies, as a generating function of the refined BPS invariants on the one hand and as thePoincar´e polynomial of the HOMFLY homology on the other. Analogous constructions for coloredHOMFLY and Kauffman homologies were developed in [GS] and [GW], respectively. Here, weincorporate the exceptional Lie algebra e and its 27-dimensional representation with (minuscule)highest weight ω .Exceptional Lie algebras pose a number of unique challenges. For one, they are singular inthe sense that they do not belong to infinite families in any obvious way. Thus, we are missinga natural notion of “stabilization,” which helps the identification of gradings/differentials in theclassical cases.In [CE], the authors consider stabilization for the Deligne-Gross “exceptional series.” However,this is a fundamentally different phenomenon than considered here, as their examples containnegative coefficients. It is an interesting question, relegated to future research, whether the approachin [CE] is compatible with the approach here.We also face a more technical/computatational challenge. Even the ordinary (quantum group)knot invariants for e have not been explicitly computed in the literature. The author R.E. hascomputed them for the cases considered here (unpublished) and verified their coincidence with theDAHA-Jones polynomials upon t (cid:55)→ q . Furthermore, no corresponding homology theory has beenformally defined.We manage to overcome these obstacles by applying the technique of differentials from [DGR, R]to the DAHA-Jones polynomials, q, t -counterparts of quantum knot invariants defined in [C4]. Thiscombination is sufficiently powerful to overcome all obstacles. Here, we propose new invariants, the hyperpolynomials , for e , torus knot homologies, as well as produce some explicit examples.3.1.1. Notation and conventions.
We will use two sets of conventions in this paper: the standardDAHA 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 inQG conventions ( q, t, u ). To change DAHA → QG, we apply the “grading change” isomorphism:(3.1) a (cid:55)→ ut − , q (cid:55)→ qt , t (cid:55)→ q. Even though q, t are used in both sets of conventions, whether we are referring to DAHA or QGwill be contextually clear.Furthermore, for a given knot, polynomials in QG conventions are usually associated to a Liealgebra g and a representation ( g -module) V . Polynomials in DAHA conventions are (equivalently)associated to a root system R and a (dominant) weight b ∈ P + . The correspondence between g and R is via the classification of complex, semisimple Lie algebras, and b is the highest weight for V , as labeled in [B]. XCEPTIONAL KNOT HOMOLOGY 17
Now, in QG-conventions, our hyperpolynomials are Poincar´e polynomials for a (hypothetical)triply-graded vector space:(3.2) H e , ( K ; q, t, u ) := (cid:88) i,j,k q i t j u k dim H e , i,j,k ( K ) . The usual two-variable Poincar´e polynomials (0.2) are returned upon setting u = 1:(3.3) P e , ( K ; q, t ) := H e , ( K ; q, t, , and we have, upon taking the graded Euler characteristic with respect to t ,(3.4) P e , ( K ; q ) = P e , ( K ; q, − , i.e. these “categorify” the quantum knot invariants (0.1) for e , .This story may be translated into DAHA conventions. In light of (3.1), we may also write thehyperpolynomials in DAHA conventions:(3.5) HD E r,s ( ω ; q, t, a ) := (cid:88) i,j,k q j + k t i − j + k a k dim H e , i,j,k ( T r,s ) . for the same vector space as in (3.2). Though we do not consider a DAHA analogue of P e , here,we may obtain the DAHA-Jones polynomial by taking the graded Euler characteristic with respectto a :(3.6) (cid:102) JD E r,s ( ω ; q, t ) = HD E r,s ( ω ; q, t, − . Recall that the DAHA-Jones polynomials are t -refinements of the QG knot invariants. They are(conjecturally) related by setting t (cid:55)→ q :(3.7) P e , ( T r,s ; q ) = (cid:102) JD E r,s ( ω ; q, q ) . 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 followingcommutative diagram.(3.8) DAHA HD a = − . (cid:47) (cid:47) (cid:79) (cid:79) (3 . (cid:15) (cid:15) (cid:102) JD t (cid:55)→ q (3 . (cid:15) (cid:15) H i,j,k (3 . (cid:57) (cid:57) (3 . (cid:37) (cid:37) QG H u =1(3 . (cid:47) (cid:47) P t = − . (cid:47) (cid:47) P Torus knots.
Presently, our approach is confined to the torus knots and links for which theDAHA-Jones polynomials are defined. The reason for this limitation is algebraic from the DAHApoint of view. Here we will shed some light on it geometrically and physically.
In the BPS framework, something special happens when K = T r,s is a torus knot. Then, the five-brane theory in (1.24) has an extra R -symmetry U (1) R that acts on S leaving the knot K = T r,s and, hence, the Lagrangian L K ⊂ T ∗ S invariant. Following [AS], we denote the quantum numbercorresponding to this symmetry by S R , and also introduce the generating function, cf. (1.21):(3.9) H g ,V ( K ; q, t, u ) := Tr H BPS q P t F u S R . that “counts” refined BPS states in the setup (1.24).From the perspective of [ORS], which is related to the DAHA approach, this extra variable /grading comes from the symmetry of the algebraic curve,(3.10) x r = y s , whose intersection with a unit sphere in R ∼ = C defines a ( r, s ) torus knot T r,s .In either case, the origin of the extra grading (resp. variable u ) has nothing to do with the choiceof homology (Khovanov, colored HOMFLY, or other); it simply comes from a very special choiceof the knot (link) and exists only for torus knots and links.As a result, what for a generic knot K might be a doubly-graded homology H g ,Vi,j ( K ) for torus knotbecomes a triply-graded homology H g ,Vi,j,k ( K ), with an extra u -grading. Likewise, what normallywould be a triply-graded (say, HOMFLY or Kauffman) homology, for a torus knot K = T r,s becomesa quadruply-graded homology H g ,Vi,j,k,(cid:96) ( T r,s ), c.f. [GGS].3.1.3. Hyper-lift.
We wish to elevate the two-variable DAHA-Jones polynomial (cid:102) JD E r,s ( ω ; q, t ), whichin general has both positive and negative coefficients, to a three-variable hyperpolynomial HD r,sE ( ω ; q, t, a )with only positive coefficients.As in (3.5), this “upgraded” polynomial will be the Poincar´e polynomial of a triply-graded vectorspace H e , i,j,k ( T r,s ), accounting for its positive coefficients. As in (3.6), it is related to (cid:102) JD E r,s by takingthe graded Euler characteristic with respect to the k -grading (resp. variable a ):(3.11) (cid:102) JD E r,s ( ω ; q, t ) = HD r,sE ( ω ; q, t, − . Note that we are here constructing the polynomial HD r,sE ( ω ) whose constituent monomials encodethe graded dimensions of the irreducible components of the vector space H E ,r,si,j,k . We are notconstructing this vector space itself.Of course, there will be many polynomials HD r,sE ( ω ) that satisfy only the aforementioned prop-erties. We will define ours intelligently so that it is uniquely determined and so that like theHOMFLY-PT (“superpolynomial”) and Kauffman homologies — which, respectively, unify sl N and so N invariants — our “hyperpolynomial” will unify the ( e , )-invariant with invariants associatedto “smaller” algebras and representations ( g , V ).3.1.4. Differentials and specializations.
This unification with other ( g , V )-colored invariants is real-ized via certain (conjectural) spectral sequences on H e , ∗ induced by deformations of the potential W E , (cid:59) W g ,V , which are studied in section 4. With the additional assumption that these spectralsequences converge on its second page, such deformations gives rise to differentials d g ,V such that XCEPTIONAL KNOT HOMOLOGY 19 the homology:(3.12) H ∗ ( H e , ∗ , d g ,V ) ∼ = H g ,V ∗ . Practically speaking, suppose that such a differential d g ,V exists (= d R,b in DAHA conventions),and that its ( q, t, a )-degree is ( α, β, γ ). Then each monomial term in HD r,sE ( ω ) will participate inexactly one of two types of direct summands in the chain complex ( H e , ∗ , d R,b ):(3.13) 0 d −→ q i t j a k d −→ , (3.14) 0 d −→ q i t j a k ∼ = −→ q i + α t j + β a k + γ d −→ . Observe that we can re-express this as a decomposition:(3.15) HD r,sE ( ω ) = (cid:103) HD R ( b ) + (1 + q α t β a γ ) Q ( q, t, a ) , where (cid:103) HD R ( b ) is related to (cid:102) JD R ( b ) by the specialization:(3.16) HD r,sE ( ω ; a = − q − αγ t − βγ ) = (cid:103) HD R ( b ; a = − q − αγ t − βγ ) = (cid:102) JD R ( b )which subsumes the differential d R,b , realized by setting (1 + q α t β a γ ) = 0. Note that since thesepolynomials always have integer exponents (corresponding to integer gradings of a vector space),we will always be able to define the a -grading in such a way that γ divides α and β .To restore the a -grading to (cid:102) JD E ( ω ), we play this game in reverse. On the q, t -level, we have adecomposition:(3.17) (cid:102) JD E ( ω ) = (cid:102) JD R ( b ) + (1 ± q α t β ) Q ( q, t ) . Since many of the polynomials (cid:102) JD R ( b ) are known, we can hope to use this structure to recover the a -gradings of specific generators as well as the a -degrees of the d R,b . If we can do this for sufficientlymany (
R, b ), we will obtain enough constraints (specializations) to uniquely define the (relative) a -grading in HD r,sE ( ω ).3.1.5. Uniqueness.
Suppose that we have defined HD by some (possibly infinite) set of differen-tials/specializations S := { ( R, b, α, β, γ ) } , each of the form (3.15) with the same (cid:103) HD R ( b ). If two hy-perpolynomials HD , HD each satisfy all of the specializations S , then evidently HD − HD ∈ I S ,where(3.18) I S := (cid:89) S (cid:16) q α t β a γ (cid:17) is an ideal in R := Z [[ q, t, a ]]. Then HD corresponds to a unique coset [ HD ] ∈ R/I S .If S is infinite, then we may choose a distinguished representative of [ HD ], i.e. the only onewith finitely many terms. This is precisely the situation when considering superpolynomials andhyperpolynomials for the classical series of Lie algebras.When S is finite, there is also a distinguished representative. Since HD is required to have positive coefficients, we may simply require that it is minimal in [ HD ] with respect to that property, i.e. ithas the minimum number of terms. Indeed, suppose HD (cid:54) = HD are minimal, and write HD − HD = F · (cid:81) S (1 + q α t β a γ ) ∈ I S for some F ∈ R . Since the HD i both have positive coefficients, we may write F = F − F , whereeach F i has only positive coefficients. Then clearly the monomials in F i · (cid:81) S (1 + q α t β a γ ) are allmonomials in HD i , and since these belong to I S , they cancel in every specialization in S . Then(3.19) HD (cid:48) i := HD i − F i · (cid:89) S (1 + q α t β a γ )is a new polynomial with positive coefficients, fewer terms, and which satisfies all of the specializa-tions S . This contradicts the assumed minimality of HD i .Restricting ourselves to these distinguished representatives, the uniqueness of our HD dependson the uniqueness of the (cid:103) HD R ( b ) chosen simultaneously for ( R, b ) ∈ S . As we will see below, thisis manifest in all cases considered.3.2. E -hyperpolynomials. In the standard knot theory (QG) conventions, our main proposalfor H e , is based on the following (finite) set of differentials/specializations:(3.20) g , V H e , ( u = 1 , t =?) = P g ,V deg( d g ,V ) e , − , − , d , − q (4 , − , a , − q (5 , − , − q (8 , − , − q − (13 , , (cid:103) HD R ( b ) in (3.15) — as wellas its variant in the QG conventions — is a single monomial.We construct three explicit examples, for T , , T , , and T , torus knots, which are also knownas the , , and knots, respectively. The result looks as follows:(3.21) H e , ( ) = 1 + q t + q t + q tu + q tu + q t + q t u + q t u + q t u (3.22) H e , ( ) = q t + q t + q tu + q tu + q t + q t + q t + q t u + 2 q t u + q t u + q t u + q t + q t + q t u + 2 q t u + q t u + q t u + q t u + q t + q t u + q t u + q t u , (3.23) H e , ( ) = q t + q t + q tu + q tu + q t + q t + q t + q t + q t + q t u + q t u + q t u + 2 q t u + q t u + q t u + q t + q t + q t + q t + q t + q t u + q t u + q t + 3 q t u + 2 q t u + 2 q t u + 2 q t u + q t u + q t u + 2 q t u + q t u + q t u + q t u + q t + q t + q t + q t + q t + q t u + q t u + q t u + q t + q t u +3 q t u +2 q t u +2 q t u +2 q t u + q t u +2 q t u + q t u + q t u + XCEPTIONAL KNOT HOMOLOGY 21 q t u + q t u + q t u + q t u + q t u + q t u + q t + q t + q t + q t u + q t u + q t + q t u + 3 q t u + q t u + q t u + 2 q t u + 2 q t u + 2 q t u + 2 q t u + q t u + q t u + 3 q t u + q t u + q t u + q t u + q t u + q t u + q t u + q t u + q t + q t u + q t u + q t u + q t u + q t u + q t u +2 q t u + q t u + q t u + q t u + q t u + q t u + q t u + q t u . Spectral sequence diagrams, which reveal the structure of the proposed differentials, are includedfor these examples in Appendix B.3.3.
Computations with DAHA-Jones polynomials.
Here we demonstrate explicitly how theDAHA-Jones polynomials are combined with the theory of differentials to produce our examples.First, we rewrite our proposal in DAHA conventions:(3.24)
R, b HD E r,s ( ω ; a =?) = (cid:102) JD R ( b ) deg( d R,b ) E , ω − , , D , ω − t − (0 , , A , ω − t − (0 , , − t − (0 , , − q − t − (1 , , H e , before the transformation (3.1). Now we consider eachof our three examples individually.3.3.1. The Trefoil T , . The DAHA-Jones ( E , ω ) polynomial for the trefoil is(3.25) (cid:102) JD E , ( ω ; q, t ) = 1 + qt + qt − qt − qt + q t − q t − q t + q t . To elevate this to a Poincar´e polynomial with positive coefficients, we introduce an extra a -grading.For now this will only be a Z / Z -grading ( a or a ) compatible with the specialization a = − HD E , ( ω ) = a + qta + qt a + qt a + qt a + q t a + q t a + q t a + q t a . Now we would like to lift this Z / Z -grading to a genuine Z -grading, for which we use thedifferential structure outlined above. Fortunately, this case is resolved rather easily by consideringthe ( D n , ω ) DAHA-Jones polynomial:(3.27) (cid:102) JD D n , ( ω ; q, t ) = 1 + qt + qt n − − qt n − qt n − + q t n − − q t n − − q t n − + q t n − , which has the same dimension as (cid:102) JD E , , so we can completely restore the a -grading by understandingjust a single differential to some ( D n , ω ), if one exists.Indeed, such a differential to ( D , ω ) is indicated by the expression:(3.28) HD E , ( ω ) = a + qta + qt a + qt a + qt a + q t a + q t a + q t a + q t a + (1 + t a )( qt a + qt a + q t a + q t a + q t a + q t a ) . Observe that the a -grading of this differential must be 1 if the corresponding specialization is tocontain only integer powers of t . Thus, the a -grading of a generator corresponds to the number ofcanceling pairs of terms required to fit that generator into the expression above. For example, thegenerator qt a is realized in (3.28) as:(3.29) qt a = qt a + (1 + t a ) qt a , so its a -grading is 1. However, the generator q t a is realized in (3.28) as:(3.30) q t a = q t a + (1 + t a )( q t a + q t a ) , so its a -grading is 2. Overall, we restore the a -grading as a Z -grading:(3.31) HD E , ( ω ) = 1 + qt + qt + qt a + qt a + q t + q t a + q t a + q t a . Observe that, as desired, we so far have the following specializations which determine the a -grading: HD E , ( ω ; a = −
1) = (cid:102) JD E , ( ω ) , (3.32) HD E , ( ω ; a = − t − ) = (cid:102) JD D , ( ω ) . (3.33)We also find the two canceling differentials: HD E , ( ω ; a = − t − ) = 1 , (3.34) HD E , ( ω ; a = − q − t − ) = q t , (3.35)as well as the differential to ( A , ω ):(3.36) HD E , ( ω ; a = − t − ) = (cid:102) JD A , ( ω ) . The Torus Knot T , . We repeat the above construction for T , and restore the a -gradingto (cid:102) JD E , ( ω ) in a way that includes all of the same structure. We have the DAHA-Jones ( E , ω )polynomial for T , :(3.37) (cid:102) JD E , ( ω ; q, t ) = qt + qt − qt − qt + q t + q t + q t − q t − q t − q t + q t + q t + q t − q t − q t − q t + q t + q t + q t − q t − q t + q t . As above, we introduce a mod-2 grading compatible with the specialization a = − HD E , ( ω ) = a + qta + qt a + qt a + qt a + q t a + q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a . The D DAHA-Jones is:(3.39) (cid:102) JD D , ( ω ; q, t ) =1 + qt + qt − qt − qt + q t + q t − q t + q t − q t − q t + q t + q t − q t + q t − q t + q t − q t + q t + q t − q t − q t + q t , XCEPTIONAL KNOT HOMOLOGY 23 which again has the same dimension as (cid:102) JD E , , so we can restore the a -grading in the same manner:(3.40) HD E , ( ω ) =1 + qt + qt + qt a + qt a + q t + q t + q t + q t a + 2 q t a + q t a + q t a + q t + q t + q t a + 2 q t a + q t a + q t a + q t a + q t + q t a + q t a + q t a . Observe that, as with the trefoil, we have specializations: HD E , ( ω ; a = −
1) = (cid:102) JD E , ( ω ) , (3.41) HD E , ( ω ; a = − t − ) = (cid:102) JD D , ( ω ) , (3.42) HD E , ( ω ; a = − t − ) = 1 , (3.43) HD E , ( ω ; a = − q − t − ) = q t , (3.44) HD E , ( ω ; a = − t − ) = (cid:102) JD A , ( ω ) . (3.45)3.3.3. The Torus Knot T , . We have the DAHA-Jones ( E , ω ) polynomial for T , :(3.46) (cid:102) JD E , ( ω ; q, t ) = qt + qt − qt − qt + q t + q t + q t + q t + q t − q t − q t − q t − q t − q t + q t + q t + q t + q t + q t + q t − q t − q t + q t − q t − q t − q t − q t + q t − q t + 2 q t + q t + q t + q t + q t + q t + q t + q t − q t − q t − q t − q t − q t − q t + 2 q t + q t − q t + 3 q t + q t + q t + q t − q t − q t + q t + q t − q t − q t − q t + 2 q t +3 q t + q t − q t − q t + q t − q t − q t + q t − q t +2 q t − q t + q t − q t − q t + q t , and the D DAHA-Jones is:(3.47) (cid:102) JD D , ( ω ; q, t ) = qt + qt − qt − qt + q t + q t + q t − q t − q t − q t + q t + q t + q t − q t + q t − q t − q t − q t + q t + q t + q t − q t − q t + q t − q t + q t . From the outset it is apparent that these do not have the same dimension, so the same approachwill be less effective. However, we can try to assign a monomial in (cid:102) JD E , ( ω ) to each monomial in (cid:102) JD D , ( ω ) so that they coincide in the specialization a = − t − . That is, we consider the followingsubset of HD E ( ω ):(3.48) HD D /E = a + qta + qt a + qt a + qt a + q ta + q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a + q t a +2 q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a , which should specialize to (cid:102) JD D , , and thus lifts to:(3.49) HD D /E = qt + qt + qt a + qt a + q t + q t + q t + q t a + 2 q t a + q t a + q t a + q t + q t + q t a + q t + 2 q t a + q t a + q t a + q t a + q t a + q t + q t a + q t a + q t a + q t a + q t a . Now we turn our eye to the complementary subset:(3.50) HD E \ D = q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a +2 q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a + 2 q t a + 2 q t a + q t a + q t a + 2 q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + 3 q t a + 2 q t a + 2 q t a + 3 q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a . We can use the degrees of the differentials (now known) to restore the a -grading on these generators.For example, q t ∈ HD D /E and q t a ∈ HD E \ D should cancel in the differential of degree(0 , , a -degree q t a on that generator. Carrying this out fully, we obtain:(3.51) HD E \ D = q t + q t + q t a + q t a + q t + q t + q t a + q t + q t a + q t a + q t a + 2 q t a + q t a + q t a + 2 q t a + q t a + q t + q t + q t + q t a + 2 q t a + 2 q t a + q t a + q t a + 2 q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a + q t + q t + q t a + 3 q t a + 2 q t a +2 q t a + 3 q t a + q t a + q t a + q t a + q t + q t a + q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a . Finally, observe that some generators that should cancel in certain specializations do not. Forexample, q t should cancel in the differential of degree (0 , , q t a . Taking alldifferentials into account, we add the generators: { q t , q t a, q t a, q t a , q t a, q t , q t , q t a , q t ,q t a , q t a , q t a, q t a, q t a, q t a , q t a, q t a , (3.52) q t a , q t aq t a, q t a , q t a , q t a , q t a } , and take the sum HD D /E + HD E \ D + (3.52) to obtain:(3.53) HD E , ( ω ) = qt + qt + qt a + qt a + q t + q t + q t + q t + q t + q t a + q t a + q t a + 2 q t a + q t a + q t a + q t + q t + q t + q t + q t + q t a + q t a + q t + 3 q t a + 2 q t a + 2 q t a + 2 q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t + q t + q t + q t + q t + q t a + q t a + q t a + q t + q t a + 3 q t a + 2 q t a + 2 q t a + 2 q t a + q t a + 2 q t a + q t a + q t a +3 q t a + q t a + q t a + q t a + q t a + q t a + q t + q t + q t + q t a + q t a + q t + XCEPTIONAL KNOT HOMOLOGY 25 q t a + 3 q t a + q t a + q t a + 2 q t a + 2 q t a + 2 q t a + 2 q t a + q t a + q t a + 3 q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t + q t a + q t a + q t a + q t a + q t a + q t a + 2 q t a + q t a + q t a + q t a + q t a + q t a + q t a + q t a , and verify that it satisfies: HD E , ( ω ; a = −
1) = (cid:102) JD E , ( ω ) , (3.54) HD E , ( ω ; a = − t − ) = (cid:102) JD D , ( ω ) , (3.55) HD E , ( ω ; a = − t − ) = 1 , (3.56) HD E , ( ω ; a = − q − t − ) = q t , (3.57) HD E , ( ω ; a = − t − ) = (cid:102) JD A , ( ω ) . (3.58)3.4. Further properties.
We observe that our hyperpolynomials exhibit a number of potentiallymeaningful structures beyond their defining specializations/differentials.3.4.1.
Dimensions.
First, observe that(3.59) HD E r,s ( ω ; q, ± , a ) = HD Ar,s ( ω ; q, ± , a )in all examples considered, in spite of the fact that the weight ω for A n is non-minuscule. Theserelations generalize the special evaluations at t = 1 of DAHA-Jones polynomials and DAHA-superpolynomials. In particular, using the evaluation and super-duality theorems from [C4], equa-tion (3.59) implies that(3.60) HD E r,s ( ω ; q, , a ) = (cid:0) HD Ar,s ( ω ; q, , a ) (cid:1) . In turn, we see that the dimensions (3.61) dim HD E r,s := HD E r,s ( ω ; 1 , , T , , T , , T , are 9 , , refined exponential growth [GS, GGS] for the exceptionalgroups.3.4.2. Hat symmetry.
We also have a “hat symmetry” corresponding to the involution of the Dynkindiagram for E which sends ω (cid:55)→ ω . We define(3.62) (cid:100) HD E r,s ( ω ; q, t, a ) := HD E r,s ( ω ; q (cid:55)→ qt , t, a (cid:55)→ at − ) , which satisfies the specializations (cid:100) HD E r,s ( ω ; q, t, −
1) = (cid:102) JD E r,s ( ω ; q, t ) , (3.63) (cid:100) HD E r,s ( ω ; q, t, − t − ) = (cid:102) JD A r,s ( ω ; qt , t ) , (3.64) (cid:100) HD E r,s ( ω ; q, t, − t − ) = 1 , (3.65) (cid:100) HD E r,s ( ω ; q, t, − q − t − ) = q α t β . (3.66)3.4.3. Other evaluations.
We also have another potentially meaningful specialization of our hyper-polynomials at a = q − t − : HD E , ( ω ; a = − q − t − ) = qt − q t + q t , (3.67) HD E , ( ω ; a = − q − t − ) = q t − q t + q t − q t + q t , (3.68) HD E , ( ω ; a = − q − t − ) = q t − q t + q t − q t + q t − q t + q t (3.69) − q t − q t + q t . We do not recognize the resulting polynomials. However, observe the significant reduction in thenumber of terms, as well as their regularity.4.
Adjacency tree of the corank-2 singularity Z , In the previous section, we encountered several “exceptional” differentials that relate homologicalinvariants of knots colored by representations of exceptional groups to knot homologies associatedwith classical groups. In this section we explain the origin of such differentials.There are two general ways to predict a priori the structure of the differentials, both of whichare rooted in physics. One approach [GS] involves analysis of the spectrum (1.21) of BPS states(a.k.a. Q -cohomology) and how it changes when one varies stability parameters, such as theK¨ahler modulus (1.7). The second approach [Go] is based on deformations of the Landau-Ginzburgpotential, which for the 27-dimensional representation of g = e has the form [GW](4.1) W E , = z − z z + z z . In general (and in every physics-based approach to knot homology), homology of the unknot canbe represented as a Q -cohomology, i.e., the space of Q -closed but not Q -exact states (called BPSstates) in a two-dimensional theory on a cylinder, R × (unknot) = R × S . In some cases, this two-dimensional theory admits a Landau-Ginzburg description, which for certain Lie algebras g andrepresentations V has been identified in [GW]. In this approach, spectral sequences and differentialscorrespond to relevant deformations and RG flows of the two-dimensional “unknot theory” which,in the Landau-Ginzburg description, simply manifest as deformations of the potential.Therefore, in our present problem we need to explore deformations of the potential (4.1) whichcorrespond to the adjacencies of the singularity Z , . Additionally, we perform a nontrivial verifica-tion of our calculations using the adjacency of the spectra of singularities. A good general referencefor material in this section is [AGV]. XCEPTIONAL KNOT HOMOLOGY 27
Singularities and Adjacency.
A singularity is an analytic apparatus that captures the localgeometry of a holomorphic (smooth) function at a critical point. For our purposes, we will considerfunctions f : C n → C and without loss of generality, critical points at 0 ∈ C n .Let O n be the space of all germs at 0 ∈ C n of holomorphic functions f : C n → C . Then thegroup of germs of diffeomorphisms (biholomorphic maps) g : ( C n , → ( C n ,
0) acts on O n by g · f = f ◦ g − . The orbits of this action define equivalence classes in O n , and those classes forwhich 0 is a critical point are called singularities . Consider a class L as a subspace of O n . An l -parameter deformation of f ∈ L ⊂ O n with base Λ = C l is the germ of a smooth map F : Λ → O n such that F (0) = f .If L is contained in the closure of some other subspace, L ⊂ ¯ K ⊂ O n , then an infinitesimalneighborhood of every f ∈ L ⊂ O n intersects K nontrivially. This geometric notion can bereformulated equivalently in terms of deformations and gives rise to the concept of adjacency. Thatis, suppose that every function f ∈ L can be transformed to a function in the class K by anarbitrarily small deformation. Here the “size” of a deformation is a restriction on λ ∈ Λ, inducedby the standard metric on C l . In this case, we say that the singularity classes L , K are adjacent ,written L → K .4.1.1. Versal deformations.
Here we aim to find the adjacencies to the specific class Z , , that is theclasses K such that Z , → K . We go about this by considering a specific type of deformation.A deformation F : Λ → O n of f is versal if every deformation of f is equivalent to one induced(by change of base Λ) from F . If, in addition, Λ has the smallest possible dimension, F is said tobe miniversal , i.e., “minimal and universal.”We can construct an explicit miniversal deformation of f ∈ L as follows. Let g t be a path ofdiffeomorphisms of ( C n ,
0) such that g is the identity. Then the tangent space T f L consists ofelements of the form(4.2) ∂∂t ( f ◦ g t ) | t =0 = n (cid:88) i =1 ∂f∂z i · ∂g i ∂t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =0 . In other words, the partial derivatives of f form an O n -linear basis for T f L , motivating the followingimportant invariants.Let I ∇ f ⊂ O n be the gradient ideal, generated by the partial derivatives of f . Then we definethe local algebra A f := O n /I ∇ f and its multiplicity or Milnor number µ := dim A f , which are bothinvariants of the singularity L .Then if { ϕ k } is a monomial basis for A f , we can define a miniversal deformation:(4.3) F ( λ ) = f + µ (cid:88) k =1 λ k ϕ k . Indeed, the graph of this deformation is a linear subspace of O n which is centered at the germ f ∈ L and is transversal to its orbit. In particular, this subspace will necessarily intersect everyclass adjacent to L . To determine these adjacent classes, we restrict to arbitrarily small (cid:15) ∈ Λ anduse Arnold’s algorithm [A1] to classify the possible F ( (cid:15) ). Nonsingular fibers and monodromy.
Let f : C n → C be a germ with (isolated) criticalpoint at 0 ∈ C n of multiplicity µ and critical value f (0) = 0. Let U be a small ball about 0 ∈ C n and B be a small ball about 0 ∈ C . If the radii of these balls are sufficiently small, the followingholds [Mi]. Theorem 4.1.
For b ∈ B (cid:48) := B \{ } , the level set X b = f − ( b ) ∩ U is a nonsingular hypersurface,homotopy equivalent to ∨ µ S n − . The level set X = f − (0) ∩ U is nonsingular away from . Then f : X (cid:48) → B (cid:48) (where X (cid:48) := f − ( B (cid:48) ) ∩ U ) is a locally trivial fibration with fiber X b (cid:39)∨ µ S n − . Suppose b ∈ ∂B is a noncritical value of f , and let [ γ ] ∈ π ( B (cid:48) , b ) ∼ = Z . Then γ ( t ) liftsto a continuous family of maps h t : X b → X t which can be chosen so that h is the identity on X b and h = h is the identity on ∂X b = f − ( b ) ∩ ∂U .The map h : X b → X b is the monodromy of γ . The induced map on homology,(4.4) h ∗ : H n − ( X b ) → H n − ( X b ) , is the corresponding monodromy operator , which is well-defined on the class [ γ ]. If, in addition,[ γ ] ∈ π ( B (cid:48) , b ) is a counterclockwise generator, h ∗ is called the classical-monodromy operator .4.2.1. Vanishing cohomology.
Observe that the (reduced) integral [co]homology is nonzero only indimension n −
1, where H n − ( X b ) ∼ = Z µ . We construct a distinguished basis for this homologygroup by first considering the simple case where f has a nondegenerate critical point of multiplicity µ = 1.The Morse lemma tells us that in some neighborhood of 0 ∈ C n , there is a coordinate system inwhich f ( (cid:126)z ) = z + · · · + z n . In this coordinate system, let S n − = { (cid:126)z : (cid:107) (cid:126)z (cid:107) = 1 , Im( z i ) = 0 } andlet ϕ : [0 , → B be a path with ϕ (0) = b and ϕ (1) = 0. Then the family of spheres,(4.5) S t = (cid:112) ϕ ( t ) S n − ⊂ X ϕ ( t ) , depends continuously on the parameter t and vanishes to the singular point S = 0 ∈ X . Thesphere S = √ b S n − corresponds to a homology class ∆ ∈ H n − ( X b ), called a vanishing cycle .In the more general case that f has a degenerate critical point of arbitrary multiplicity µ ,one can slightly perturb f into a function f (cid:15) = f + (cid:15)g with µ nondegenerate critical points ina small neighborhood of 0 ∈ C n , having distinct critical values a i . Now consider a system ofpaths ϕ , . . . , ϕ µ with ϕ i (0) = b and ϕ i (1) = a i . Suppose that these paths satisfy the followingconditions:(1) The loops formed by traversing ϕ i , followed by a small counterclockwise loop around a i ,followed by ϕ − i generate π ( B (cid:48) , b )(2) The paths ϕ i do not intersect themselves and intersect each other only at b for t = 0(3) The paths are indexed clockwise in arg ϕ i ( (cid:15) )Then, as above, each path ϕ i determines a distinct vanishing cycle ∆ i ∈ H n − ( X b ), and the set { ∆ , . . . , ∆ µ } form a distinguished basis of vanishing cycles for the homology H n − ( X b ) ∼ = Z µ .4.3. Mixed Hodge structure in the vanishing cohomology.
For f : X (cid:48) → B (cid:48) as in Theo-rem 4.1, the µ -dimensional complex vector bundle π ∗ f : H ∗ f → B (cid:48) , whose fibers are the complex[co]homology groups H n − ( X b ; C ), is called the vanishing [co]homology bundle of the singularity XCEPTIONAL KNOT HOMOLOGY 29 f . There is a natural connection ∇ in the vanishing [co]homology bundle, called the Gauss-Maninconnection , which is defined by covariant derivation ∇ b along the holomorphic vector field ∂∂b onthe base B (cid:48) .We would like to define a mixed Hodge structure in the vanishing cohomology bundle and soreview the relevant definitions. Suppose we have an integer lattice H Z in a real vector space H R = H Z ⊗ Z R . Let H = H Z ⊗ Z C be its complexification. Then for k ∈ Z , a pure Hodge structureof weight k on H is a decomposition:(4.6) H = (cid:77) p + q = k H p,q , into complex subspaces satisfying H p,q = H q,p , where the bar denotes complex conjugation in C .Equivalently, we may specify a Hodge structure by a Hodge filtration : a finite, decreasing filtra-tion F p on H satisfying F p ⊕ F p +1 = H . Indeed, from a Hodge filtration, one can recover a Hodgestructure by H p,q = F p ∩ F q , and from a Hodge structure, one can recover a Hodge filtration by F p = ⊕ i ≥ p H i,k − i . We generalize these notions to a mixed Hodge structure on H , specified by(1) A weight filtration : a finite, increasing filtration W k on H which is the complexification ofan increasing filtration on H Z ⊗ Z Q ,(2) A Hodge filtration : a finite, decreasing filtration F p on H ,such that for each k , the filtration,(4.7) F p gr Wk H := ( F p ∩ W k + W k − ) /W k − , satisfies F p gr Wk H ⊕ F k − p +1 gr Wk H = gr Wk H . That is, F p gr Wk H induces a pure Hodge structure ofweight k on gr Wk H := W k /W k − .The vanishing cohomology is obtained as the complexification of the integral cohomology of thenonsingular fibers X b . So to define a mixed Hodge structure in the vanishing cohomology, it re-mains to specify the relevant weight and Hodge filtrations there. We follow the construction of[V1, V2, V3].4.3.1. Hodge filtration.
To obtain a Hodge filtration, first consider a holomorphic ( n − ω defined in a neighborhood of 0 ∈ C n . Since X b is a complex ( n − ω b = ω | X b represents a cohomology class [ ω b ] ∈ H n − ( X b , C ) for all b ∈ B (cid:48) . That is, ω defines aglobal section s ω : B (cid:48) → H ∗ f , b (cid:55)→ [ ω b ] of the vanishing cohomology bundle.In the neighborhood of every nonsingular manifold X b , there exists a holomorphic ( n − ω/df , with the property that ω = df ∧ ω/df in that neighborhood. As above, the restriction ω/df b = ω/df | X b , called the residue form , represents a cohomology class [ ω/df b ] ∈ H n − ( X b , C ) anddefines a global section σ ω : B (cid:48) → H ∗ f , b (cid:55)→ [ ω/df b ] of the vanishing cohomology bundle.The section σ ω is called a geometric section . For a set of µ forms that do not satisfy a complexanalytic relation, the set of their geometric sections trivalizes the vanishing cohomology bundle,i.e., the corresponding residue forms are a basis in each fiber.The above sections are holomorphic, meaning that if δ b is a cycle in the (integer) homologyof the fiber which depends continuously on b , (i.e., is covariantly constant via the Gauss-Manin connection), then the map σ ω δ : b (cid:55)→ (cid:82) δ b σ ω ( b ) is a holomorphic map B (cid:48) → C . We consider anasymptotic expansion of such a map around zero.For example, in the simple case (4.5) of a nondegenerate critical point, one can easily see that(4.8) s ω S ( b ) = (cid:90) S b s ω ( b ) = (cid:90) B b ds ω ( b ) = cb n/ + · · · .S b is the vanishing sphere, B b is its interior, and the expansion is proportional to vol B b and dω | .For general f with a (possibly degenerate) critical point at 0, we can take a set of forms ω , . . . , ω µ such that their geometric sections trivialize the vanishing cohomology bundle. Then analysis of thePicard-Fuchs equations of these geometric sections yields the following theorem. Theorem 4.2.
Let δ b be a continuous family of vanishing cycles over the sector θ < arg b < θ in B (cid:48) . Let σ ω be a section of the vanishing cohomology. Then the corresponding integral admits anasymptotic expansion: (4.9) σ ω δ ( b ) = (cid:90) δ b ω/df b = (cid:88) k,α T k,α b α (log b ) k k ! , which converges for b sufficiently close to 0. The numbers e πiα are the eigenvalues of the classicalmonodromy operator. If we fix ω , the coefficients T k,α do not depend on b , but they do depend linearly on the section δ ,and so determine sections τ ωk,α of the vanishing cohomology bundle via the pairing (cid:104) τ ωk,α , δ (cid:105) = T k,α between homology and cohomology. Thus, we can rewrite the asymptotic expansion (4.9) as aseries expansion of the geometric section:(4.10) σ ω = (cid:88) k,α τ ωk,α b α (ln b ) k k !This expansion induces a filtration of the vanishing cohomology as follows. Define(4.11) α ( ω ) := min { α : ∃ k ≥ τ ωk,α (cid:54) = 0 } Given a geometric section σ ω , the number α ( ω ) is its order , and the corresponding expansion:(4.12) Σ ω := (cid:88) k τ ωk,α ( ω ) b α ( ω ) (ln b ) k k !is its principal part . Now define a finite, decreasing filtration of F pb of the fiber H n − ( X b ; C ) by(4.13) F pb := (cid:104) Σ ω,b : α ( ω ) ≤ n − p − (cid:105) ⊆ H n − ( X b ; C ) , and the asymptotic Hodge filtration filtration of the vanishing cohomology bundle by:(4.14) F p := (cid:91) b F pb . Weight filtration.
Suppose we have a nilpotent operator N acting on a finite-dimensionalvector space H . Then there is exactly one finite, increasing filtration W k on H which satisfies:(1) N ( W k ) ⊂ W k − (2) N k : W r + k /W r + k − ∼ = W r − k /W r − k − for all k XCEPTIONAL KNOT HOMOLOGY 31 called the weight filtration of index r of N .We obtain a weight filtration in the vanishing cohomology bundle using this construction and theclassical-monodromy operator M . As is true for any invertible linear operator, M has a Jordan-Chevalley decomposition M = M u M s into commuting unipotent and semisimple parts. Define anilpotent operator N to be the logarithm of the unipotent part:(4.15) N = (cid:88) i ( − i +1 ( M u − I ) i i . Now for each eigenvalue λ of the monodromy operator on H n − ( X b ; C ), let H λ,b be the correspond-ing root subspace. Define a filtration W k,b,λ according to the following rules:(1) If λ = 1, let W k,b,λ be the weight filtration of index n of N on H λ,s (2) If λ (cid:54) = 1, let W k,b,λ be the weight filtration of index n − N on H λ,s Now define a filtration W k,b of the fiber H n − ( X b ; C ) by:(4.16) W k,b := (cid:77) λ W k,b,λ and a filtration W k of the vanishing cohomology bundle by:(4.17) W k := (cid:91) b W k,b The subbundle W k is the weight filtration in the vanishing cohomology bundle. Now we may statethe following theorem from [V3]. Theorem 4.3.
For all k and p , the filtrations W k and F p are analytic subbundles of the vanishingcohomology bundle, which are invariant under the action of the semisimple part of the monodromyoperator. Furthermore, they specify a mixed Hodge structure in the vanishing cohomology bundle: (4.18) gr Wk H = (cid:77) p + q = k H p,q , where H p,q := F p ∩ W k /F p +1 ∩ W k + W k − . Spectrum of a singularity.
In light of Theorem 4.3, we are now in a position to define thespectrum of a singularity f ∈ K .Let f ∈ K be a singularity. If λ is an eigenvalue of the semisimple part of the classical-monodromyoperator on H p,q , one can associate to f the set of µ rational numbers:(4.19) { n − − l p λ } , where l p λ := log( λ/ πi ) and Re( l p λ ) = p. This (unordered) set of numbers is the spectrum of the singularity K .To see what the spectrum of a singularity f has to do with adjacencies to f , we first constructa fibration, analagous to the fibration f : X (cid:48) → B (cid:48) . Choose a miniversal deformation:(4.20) F ( z, λ ) = f ( z ) + µ − (cid:88) i =0 λ i ϕ i ( z ) where λ ∈ C µ and ϕ = 1. As before, we consider sufficiently small ball U about 0 ∈ C n and aanother small ball, this time Λ about 0 ∈ C µ .For λ ∈ Λ, define the level set V λ := { z ∈ U : F ( z, λ ) = 0 } and the hypersurface V := { ( z, λ ) ∈ U × Λ : F ( z, λ ) = 0 } . Let Σ ⊂ Λ be the set of values of λ for which V λ is singular, called the levelbifurcation set . Let π Λ : V → Λ be the restriction of the canonical projection, called the
Whitneymap . Finally, let Λ (cid:48) := Λ \ Σ and V (cid:48) := π − (Λ (cid:48) ). The locally trivial fibration π Λ : V (cid:48) → Λ (cid:48) withfiber V λ over λ ∈ Λ (cid:48) is the Milnor fibration of f .Observe that the fibration f : X (cid:48) → B (cid:48) can be embedded in the Milnor fibration by identifying B (cid:48) with the λ -axis in the base Λ (cid:48) (recall that ϕ = 1). Furthermore, we can repeat the constructionsoutlined above for the Milnor fibration and then ask how the spectrum varies as we vary thedeformation parameter λ in an infinitesimal neighborhood of 0. This leads to observations on the semicontinuity of the spectrum, including the following [A2]. Theorem 4.4.
Suppose that a critical point of type L has (ordered) spectrum α ≤ · · · ≤ α µ anda critical point of type L (cid:48) has spectrum α (cid:48) ≤ · · · ≤ α (cid:48) µ (cid:48) where µ (cid:48) ≤ µ . Then a necessary conditionfor the adjacency L → L (cid:48) is that the spectra be adjacent in the sense that α i ≤ α (cid:48) i . XCEPTIONAL KNOT HOMOLOGY 33
Appendix A. DAHA-Jones formulas
Type A . The formulas for (cid:102) JD A n ( b ), can be readily obtained from the following well-known type- A super-polynomials HD Ar,s ( b ; q, t, a ) upon the substitution a = − t n +1 . We will need only A here,which corresponds to a = − t : HD A , ( ω ) = 1 + aq + qt, (A.1) HD A , ( ω ) = 1 + qt + q t + a (cid:0) q + q t (cid:1) , (A.2) HD A , ( ω ) = 1 + a q + qt + q t + q t + q t + a (cid:0) q + q + q t + q t + q t (cid:1) . (A.3)The simplest colored formulas for the super-polynomials of type A , defined for ω , are knownfrom [GS, FGS] and [C4]. They play an important role for the super-polynomials of the pair( E , ω ), in spite of the fact that this weight is non-minuscule. HD A , ( ω ; q, t, a ) =(A.4) 1 + a q t + qt + qt + q t + a (cid:0) q + qt + q t + q t (cid:1) , (A.5) HD A , ( ω ; q, t, a ) =1 + qt + qt + q t + q t + q t + q t + q t + q t + a (cid:0) q + q t + q t + q t (cid:1) + a (cid:0) q + q + qt +2 q t + q t + q t + 2 q t + q t + q t + q t (cid:1) , (A.6) HD A , ( ω ; q, t, a ) = a q t + qt + q t + qt + 2 q t + q t + 2 q t + q t + 2 q t + q t + q t + q t + q t + 2 q t + q t + q t + q t + q t + q t + q t + q t + a (cid:0) q + q + q t + q t + q t + q t + q t + q t + q t + q t (cid:1) + a (cid:0) q +2 q + q + q t + q t + q t + q t + q t + 4 q t + 2 q t + 2 q t + 3 q t + q t + 3 q t + q t + 2 q t + q t + q t +2 q t + q t + q t (cid:1) + a (cid:0) q + 2 q + q + qt + q t + 2 q t + 4 q t + q t + q t + 4 q t + 2 q t + 2 q t + 4 q t + q t + q t + 4 q t + 2 q t + 2 q t + 3 q t + q t + 3 q t + 2 q t + q t + q t + q t + q t + q t (cid:1) . More specifically, we will need the values of these super-polynomials at t = 1: HD A , ( ω ; q, t = 1 , a ) = (1 + q + aq ) , (A.7) HD A , ( ω ; q, t = 1 , a ) = (1 + q + aq + q + aq ) , (A.8) HD A , ( ω ; q, t = 1 , a ) = (1 + q + aq + 2 q + 2 aq + q + 2 aq + a q ) . (A.9)For instance, the corresponding dimensions HD A ( q = 1 , t = 1 , a = 1) are 9 , , Type D . We will need the following DAHA-Jones polynomials of type D for ω (which is minus-cule): (cid:102) JD D , ( ω ; q, t ) =(A.10) 1 + qt + qt − qt − qt + q t − q t − q t + q t , (A.11) (cid:102) JD D , ( ω ; q, t ) =1 + qt + q t + qt − qt + q t − q t − qt + q t − q t + q t − q t − q t + q t + q t − q t + q t − q t + q t + q t − q t − q t + q t , (A.12) (cid:102) JD D , ( ω ; q, t ) =1 + qt + q t + qt − qt + q t − q t − qt + q t − q t + q t − q t − q t + q t + q t − q t + q t − q t + q t + q t − q t − q t + q t . We will also need the super-polynomials for the case when the last fundamental weight is takenfor D n ( n ≥ (cid:100) HD D , ( ω n ) = 1 + aqt + qt , (A.13) (cid:100) HD D , ( ω n ) = 1 + qt + q t + a (cid:0) qt + q t (cid:1) , (A.14) (cid:100) HD D , ( ω n ) = 1 + a q t + qt + q t + q t + a (cid:0) qt + q + q t + q t + q t + q t (cid:1) , (A.15)where the relevant specializations are(A.16) (cid:100) HD D ( q, t, a (cid:55)→ − t n − ) = (cid:102) JD D n ( ω n ; q, t ) . The DAHA-superpolynomials and DAHA-Jones polynomials for ω n − are identical to those for ω n .Interestingly, these super-polynomials are related to those for ( A , ω ): (cid:100) HD Dr,s ( ω n ; q, t, a ) = HD Ar,s ( ω ; q (cid:55)→ tq , t, a (cid:55)→ at ) , (A.17)so we have essentially similar “stable theories” for the pairs ( A n − , ω ) and ( D n , ω n ). Type E . We will need the DAHA-Jones polynomials for the minuscule weight ω : (cid:102) JD E , ( ω ; q, t ) =(A.18) 1 + q (cid:0) t + t − t − t (cid:1) + q (cid:0) t − t − t + t (cid:1) , (A.19) (cid:102) JD E , ( ω ; q, t ) =1 + q (cid:0) t + t − t − t (cid:1) + q (cid:0) t + t + t − t − t − t + t (cid:1) + q (cid:0) t + t − t − t − t + t + t (cid:1) + q (cid:0) t − t − t + t (cid:1) , (A.20) (cid:102) JD E , ( ω ; q, t ) = q (cid:0) t + t − t − t (cid:1) + q (cid:0) t + t + t + t + t − t − t − t − t − t + t (cid:1) + q (cid:0) t + t + t + t + t − t − t + t − t − t − t − t + t − t + 2 t + t + t + t (cid:1) + q (cid:0) t + t + t + t − t − t − t − t − t − t + 2 t + t − t + 3 t + t + t + t − t − t (cid:1) + q (cid:0) t + t − t − t − t + 2 t + 3 t + t − t − t (cid:1) + q (cid:0) t − t − t + t − t + 2 t − t + t − t − t + t (cid:1) . XCEPTIONAL KNOT HOMOLOGY 35
The next series of DAHA-Jones polynomials will be for ω (minuscule): (cid:102) JD E , ( ω ; q, t ) =(A.21) 1 + q ( t + t − t − t ) + q ( t − t − t + t ) , (A.22) (cid:102) JD E , ( ω ; q, t ) =1 + q (cid:0) t + t − t − t (cid:1) + q (cid:0) t + t − t + t − t − t + t (cid:1) + q (cid:0) t − t + t − t + t − t + t (cid:1) + q (cid:0) t − t − t + t (cid:1) , (A.23) (cid:102) JD E , ( ω ; q, t ) =1 + q (cid:0) t + t − t − t (cid:1) + q (cid:0) t + t + t − t − t − t + t (cid:1) + q (cid:0) t + t − t + t − t − t − t + t + t (cid:1) + q (cid:0) t − t − t + t − t + t (cid:1) . Appendix B. Figures
This appendix contains diagrams which depict our proposals for H e , in Section 3. We useQG-conventions; see (3.8). In particular, Figure 1 corresponds to our proposal for T , , figure 2corresponds to our proposal for T , , and figure 2 corresponds to our proposal for T , .In each figure, a monomial q i t j u k corresponds to the number k placed on the diagram in position( i, j ), i.e., with x -coordinate i and y -coordinate j . The differentials are depicted by line segmentsconnecting pairs of monomials, color-coded as follows.(B.1) g , V color deg( d g ,V ) e , – (0 , − , d , Red (4 , − , a , Yellow (5 , − , , − , , , d , ) only appears in the diagram for H e , ( T , ), that structure still exists as a specialization in the other two cases; see Section 3. q t Figure 1.
Differentials for T , X C EP T I O NA L K N O T H O M O L O G Y q t Figure 2.
Differentials for T , R O SS E LL I O T AN D S E R G E I G U K O V
10 20 30 40 q t Figure 3.
Differentials for T , Appendix C. Adjacencies and spectra
This appendix contains the adjacency tree to Z , , computed as outlined in section 4. This treedisplays only those adjacencies (“arrows”) that arise in the classification of singularities by theirjets [A1], though there are other internal adjacencies. Observe that as a direct consequence of thedefinition of adjacency, this tree is transitive in that A → B → C implies A → C .One can check this list using the adjacency of the spectra, which are also listed. There are manyways to compute the spectrum of a singularity, and we will outline one method here. Suppose f ∈ O n with Taylor expansion f = (cid:80) a k z k . Then we can take the set:(C.1) supp f = { k ∈ N n ≥ : a k (cid:54) = 0 } Now we let define a subset of R n + by:(C.2) G ( f ) = (cid:91) k ∈ supp f { k + R n + } The convex hull of G ( f ) constitutes the Newton polyhedron of f , and the union of the compactfaces of the Newton polyhedron is the Newton diagram Γ( f ) of f .A Newton diagram induces a decreasing filtration on power series as follows. If we assume thatany monomial contained in the Newton diagram is quasihomogeneous of degree 1, then each face e i ∈ Γ( f ) determines a set of weights ν i such that (cid:104) j , ν i (cid:105) = 1 for all z j ∈ e i . We can then define theNewton degree of an arbitrary monomial by:(C.3) deg z k = min i (cid:104) k , ν i (cid:105) . Then if every monomial in a power series has Newton degree greater than or equal to d , that powerseries belongs to the d th subspace of the Newton filtration.The Newton filtration also descends to forms, e.g., the Newton order of the form z k dz ∧ · · · ∧ dz n coincides with the Newton order of the monomial z k z · · · z n . Furthermore, the Newton filtrationon forms coincides with the Hodge filtration after a shift of indices, and one can show that for anappropriate set of monomials (ones whose corresponding forms trivialize the vanishing cohomologybundle), the spectrum coincides with the set of numbers:(C.4) min i (cid:104) k + , ν i (cid:105) − , for those monomials, which can often be taken to be a basis for the local algebra or, using thesymmetry of the spectrum about n −
1, a set of subdiagrammatic monomials –those z k for which k + does not belong to the interior of the Newton polyhedron. The following table lists the singularities adjacent to Z , and their normal forms, relevant defor-mations, and Milnor numbers. Singularity Normal Form ∆W /(cid:15) µZ , x y + dx y + a xy + y −− A k , 1 ≤ k ≤ x k +1 x + y k +1 kD k , 4 ≤ k ≤ x y + y k − x y + y k − kE x + y x + y E x + xy x + xy E x + y x + y J , x + bx y + y x + x y + y J ,p , 1 ≤ p ≤ x + x y + ay p x + x y + y p
10 + pE x + y + a xy x + y E x + xy + a y x + xy E x + y + a xy x + y J , x + bx y + y + axy x + x y + y J ,p , 1 ≤ p ≤ x + x y + a y p x + x y + y p
16 + pE x + y + a xy x + y E x + xy + a y x + xy E x + y + a xy x + y J , x + bx y + y + a xy x + x y + y J , x + x y + a y x + x y J , x + x y + a y x + xy + x y E x + y + a xy x + y E x + xy + a y x + xy X , x + ax y + y , a (cid:54) = 4 y + xy + x y X ,p , 1 ≤ p ≤ x + x y + ay p , a (cid:54) = 0 x y + y p pZ x y + y + axy y Z x y + xy + ax y xy Z x y + y + axy y Z , x y + dx y + axy + y x y + y Z ,p , 1 ≤ p ≤ x y + x y + a y p x y + y p
15 + pZ x y + y + a xy y Z x y + xy + a y xy Z x y + y + a xy y Z , x y + dx y + a xy + y x y + y Z ,p , 1 ≤ p ≤ x y + x y + a y p x y + y p
21 + pZ x y + y + a xy y Z x y + xy + a y xy Z x y + y + a xy y Here we have that a k := a + · · · + a k − y k − , a := 0. XCEPTIONAL KNOT HOMOLOGY 41
The following table lists the spectra of the singularities which are adjacent to Z , . Observe that,by Theorem 4.4, it supports our list of adjacencies. Spectrum Z , − , − , , , , , , , , , , , , , , , , , , , , , , , , , A k k +2 , k +2 , ··· , k k +2 D k { k − , k − , ··· , k − k − } ∪ { k − k − } E , , , , , E , , , , , , E , , , , , , , J ,p { p +6) , p +6) , ··· , p +6)6( p +6) } ∪ { p +6)6( p +6) , p +6)6( p +6) , p +6)6( p +6) } E − , , , , , , , , , , , E − , , , , , , , , , , , , E − , , , , , , , , , , , , , J ,p { p +9) , p +9) , ··· , p +17)18( p +9) } ∪ { − ( p +9)18( p +9) , p +9)18( p +9) , p +9)18( p +9) , p +9)18( p +9) , p +9)18( p +9) , p +9)18( p +9) , p +9)18( p +9) } E − , , , , , , , , , , , , , , , , , E − , , , , , , , , , , , , , , , , , , E − , , , , , , , , , , , , , , , , , , , J ,p { p +12) , p +12) , ··· , p +11)12( p +12) }∪{ − ( p +12)12( p +12) , p +12) , p +12)12( p +12) , p +12)12( p +12) , p +12)12( p +12) , p +12)12( p +12) , p +12)12( p +12) , p +12)12( p +12) , p +12)12( p +12) , p +12)12( p +12) , p +12)12( p +12) } E − , − , , , , , , , , , , , , , , , , , , , , , , E − , − , , , , , , , , , , , , , , , , , , , , , , , X ,p { p +4) , p +4) , ··· , p +4)4( p +4) } ∪ { p +44( p +4) , p +4)4( p +4) , p +4)4( p +4) , p +4)4( p +4) } Z − , , , , , , , , , , Z − , , , , , , , , , , , Z − , , , , , , , , , , , , Z ,p { p +7) , p +7) , ··· , p +13)14( p +7) } ∪ { − ( p +7)14( p +7) , p +7)14( p +7) , p +7)14( p +7) , p +7)14( p +7) , p +7)14( p +7) , p +7)14( p +7) , p +7)14( p +7) , p +7)14( p +7) } Z − , , , , , , , , , , , , , , , , Z − , , , , , , , , , , , , , , , , , Z − , , , , , , , , , , , , , , , , , , Z ,p { p +10) , p +10) , ··· , p +10)10( p +10) }∪{ − ( p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) , p +10)10( p +10) } Z − , − , , , , , , , , , , , , , , , , , , , , , Z − , − , , , , , , , , , , , , , , , , , , , , , , Z − , − , , , , , , , , , , , , , , , , , , , , , , , R O SS E LL I O T AN D S E R G E I G U K O V Adjacency tree to Z , : A A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) A (cid:111) (cid:111) D (cid:98) (cid:98) D (cid:98) (cid:98) (cid:111) (cid:111) D (cid:98) (cid:98) (cid:111) (cid:111) D (cid:98) (cid:98) (cid:111) (cid:111) D (cid:98) (cid:98) (cid:111) (cid:111) D (cid:99) (cid:99) (cid:111) (cid:111) D (cid:100) (cid:100) (cid:111) (cid:111) D (cid:100) (cid:100) (cid:111) (cid:111) D (cid:100) (cid:100) (cid:111) (cid:111) D (cid:100) (cid:100) (cid:111) (cid:111) D (cid:111) (cid:111) E (cid:89) (cid:89) (cid:98) (cid:98) E (cid:89) (cid:89) (cid:98) (cid:98) (cid:111) (cid:111) E (cid:89) (cid:89) (cid:98) (cid:98) (cid:111) (cid:111) J , (cid:105) (cid:105) J , (cid:111) (cid:111) J , (cid:111) (cid:111) J , (cid:111) (cid:111) J , (cid:111) (cid:111) J , (cid:111) (cid:111) J , (cid:111) (cid:111) J , (cid:111) (cid:111) E (cid:100) (cid:100) E (cid:100) (cid:100) (cid:111) (cid:111) E (cid:100) (cid:100) (cid:111) (cid:111) J , (cid:106) (cid:106) J , (cid:111) (cid:111) J , (cid:111) (cid:111) J , (cid:111) (cid:111) J , (cid:111) (cid:111) E (cid:100) (cid:100) E (cid:100) (cid:100) (cid:111) (cid:111) E (cid:100) (cid:100) (cid:111) (cid:111) J , (cid:105) (cid:105) J , (cid:111) (cid:111) J , (cid:111) (cid:111) E (cid:100) (cid:100) E (cid:100) (cid:100) (cid:111) (cid:111) X , (cid:85) (cid:85) X , (cid:111) (cid:111) X , (cid:111) (cid:111) X , (cid:111) (cid:111) X , (cid:111) (cid:111) X , (cid:111) (cid:111) X , (cid:111) (cid:111) X , (cid:111) (cid:111) X , (cid:111) (cid:111) X , (cid:111) (cid:111) Z (cid:100) (cid:100) Z (cid:100) (cid:100) (cid:111) (cid:111) Z (cid:100) (cid:100) (cid:111) (cid:111) Z , (cid:106) (cid:106) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z (cid:100) (cid:100) Z (cid:100) (cid:100) (cid:111) (cid:111) Z (cid:100) (cid:100) (cid:111) (cid:111) Z , (cid:105) (cid:105) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z , (cid:111) (cid:111) Z (cid:100) (cid:100) Z (cid:100) (cid:100) (cid:111) (cid:111) Z (cid:100) (cid:100) (cid:111) (cid:111) XCEPTIONAL KNOT HOMOLOGY 43
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