A Non-Abelian Generalization of the Alexander Polynomial from Quantum sl 3
AA GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROMHIGHER RANK QUANTUM GROUPS
MATTHEW HARPER
Abstract.
Murakami and Ohtsuki have shown that the Alexander polynomial is an R -matrix invariant associated with representations V ( t ) of unrolled restricted quantum sl ata fourth root of unity. In this context, the highest weight t ∈ C × of the representationdetermines the polynomial variable. For any semisimple Lie algebra g of rank n , we extendtheir construction to a link invariant ∆ g , which takes values in n -variable Laurent polyno-mials. We prove general properties of these invariants, but the focus of this paper is the case g = sl . For any knot K , evaluating ∆ sl at t = ± t = ±
1, or t = ± it − recovers theAlexander polynomial of K . We emphasize that this is not obvious from an examinationof the R -matrix and that our proof requires several representation-theoretic results. Wetabulate ∆ sl for all knots up to seven crossings along with various other examples. Inparticular, this invariant can detect mutation and is non-trivial on Whitehead doubles. Introduction
Since the introduction of the Jones polynomial, an outstanding problem in quantum topol-ogy has been to give interpretations of quantum invariants of knots and 3-manifolds in termsof invariants from classical topology. Our motivating example is the Alexander-Conwaypolynomial, realized as the quantum invariant from unrolled restricted quantum sl at aprimitive fourth root of unity. In [Mur92, Mur93, Oht02], Murakami and Ohtuski constructthe Alexander-Conway polynomial from Turaev-type [Tur88] R -matrix actions on a familyof quantum group representations. We denote these representations by V ( t ), with t ∈ C × the highest weight of this two dimensional Verma module.In contrast to the Jones polynomial, whose variable is the quantum parameter q of U q ( sl ),the variable in the Alexander-Conway polynomial is the parameter t ∈ C × of the unrolledrestricted quantum group representation V ( t ). Unlike the standard representation of U q ( sl ), V ( t ) has quantum dimension zero; therefore, the naive R -matrix invariant assigns the valueof zero to any closed tangle. Instead, using a modified trace, given by computing the invari-ant after cutting an arbitrary strand of the link, yields a nontrivial invariant.Although many higher rank quantum invariants have been defined in the literature, they arenot as well understood as quantum invariants in rank one. The HOMFLY, Kauffman, andKuperberg polynomials [Kau90, FHL +
85, Kup94] are higher rank versions of the Jones poly-nomial. The Links-Gould invariants, among others, generalize the Alexander polynomial asan R -matrix invariant from quantum supergroups [LG92, KS91, GP07]. However, a higherrank version of the Alexander polynomial from unrolled restricted quantum groups has notbeen studied and is the subject of this paper. a r X i v : . [ m a t h . QA ] A ug MATTHEW HARPER
Let h ⊆ g denote the Cartan subalgebra of a Lie algebra g of rank n . Each t = ( t , . . . , t n ) ∈ Hom( h , C ) ∼ = ( C × ) n determines a Verma module V ( t ) over the restricted quantum group,which is studied further in [Har19b]. We assume q = ζ is a primitive fourth root of unityand extend V ( t ) to a representation of the unrolled restricted quantum group. The associ-ated quantum invariant ∆ g assigns a Laurent polynomial in Z [ t ± , . . . , t ± n ] to every knot andlink. We compute ∆ g from a modified trace after coloring each component of a link L by V ( t ).This invariant is not to be confused with the multivariable Alexander polynomial. In par-ticular, the number of variables in ∆ g depends on the rank of g and not on the number ofcomponents of L . In fact, we compare ∆ sl with the Alexander polynomial and other invari-ants in Section 1.1. However, if L has m components, one can consider a modified versionof ∆ g by coloring each component of L by a distinct representation V ( t ). The resulting in-variant in mn variables is the multivariable Alexander analog of ∆ g , but we will not discussit further here.1.1. Statement of Results.
We prove that any graph automorphism of the Dynkin dia-gram of g determines a symmetry of ∆ g . In addition, if g is of type A, D, E, G, or B even ,then ∆ g is preserved under the map t (cid:55)→ − t − . Formal statements are given in Corollary6.2 and Lemma 6.3. The focus of this paper is ∆ sl , for which we prove a stronger result. Proposition 1.1.
For all links L , ∆ sl ( L )( t , t ) is a Laurent polynomial in t and t suchthat ∆ sl ( L )( t , t ) = ∆ sl ( L )( t , t ) = ∆ sl ( L )( t − , t − ) . (1)The value of ∆ sl on all prime knots up to seven crossings is tabulated in Figure 8. Wehave also computed this invariant for some higher crossing knots, allowing us to compare itwith the Alexander, Jones, and HOMFLY polynomials. These values are found in Figure 9.The HOMFLY polynomial does not distinguish the knot from , but ∆ sl does. TheJones polynomial differentiates and , but ∆ sl does not. The Jones polynomial andthe sl invariant both distinguish from ; however, the Alexander polynomial does not.We remark that the Conway knot n34 and the Kinoshita-Terasaka knot n42 have different∆ sl polynomials. Therefore, ∆ sl can detect mutation. This observation is consistent withthe following result of Morton and Cromwell [MC96]: Colorings by representations with amultiplicity-free tensor product cannot detect mutation. The polynomial invariants of n34 and n42 are determined from Figure 1 below, as explained in Section 7.In addition to n34 and n42 , untwisted Whitehead doubles of knots have Alexander poly-nomial equal to 1 [Rol03]. It follows that the Alexander module of each of these knots iszero. We take the Whitehead double of the trefoil Wh ( ) as an example. In contrast to theAlexander invariant, ∆ sl assigns a non-trivial polynomial to the mutant pair and Wh ( ),see Figures 1 and 2.The Alexander-Conway polynomial ∆ is dominated by ∆ sl on knots and it is in the followingsense that ∆ sl can be interpreted as a two-variable generalization of the classical knotinvariant. GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 3 −
44 22 − − − − − − − − − − − − n34 − − − − − n42 Figure 1.
The value of ∆ sl on the mutant pair n34 and n42 . -23 12 − Wh ( ) Figure 2.
The value of ∆ sl on the untwisted Whitehead double of . Theorem 1.2 (Reduction to the Alexander-Conway Polynomial) . Let K be a knot. Then ∆ sl ( K )( t, ±
1) = ∆ sl ( K )( ± , t ) = ∆ sl ( K )( t, ± it − ) = ∆( K )( t ) . (2) Moreover, these are the only substitutions which yield the Alexander polynomial on all knots.
Unlike the rank one case, this result cannot be deduced from the characteristic polynomialof the R -matrix for the specified parameters. As we will see, these parameter values coin-cide precisely with the values of t ∈ ( C × ) for which V ( t ) is reducible. If V ( t ) fits into ashort exact sequence with submodule V and quotient V , then for knots, ∆ sl is equal to therepresentation-theoretic average of the invariants which color the knot by V and V . TheSchur’s lemma argument which proves this only applies to knots, so Theorem 1.2 is false fora general link. To prove the invariants obtained from coloring by V and V are both relatedto the Alexander polynomial requires a separate argument. That result is stated below inTheorem 1.7.We give an example of how Theorem 1.2 does not apply to links. For a multiple componentlink L colored by a single representation, we may choose to normalize ∆ sl ( L ) by the Hopflink: ∆ sl ( ) = ( t − t − )( t − t − )( t t + t − t − ) . (3)This normalization is analogous to the factor of ( t i − t − i ) − considered when computing themulti-variable Alexander polynomial as a quantum invariant [Har19a, Jia16, Mur93, Oht02].Normalization by the factor ∆ sl ( ) does not lead to a specialization of the sl invariant of L to the Alexander polynomial. For example, let L = T , , the (4 ,
2) torus link. We see that∆( T , )( t ) = t + t − is not obtained from a “simple” evaluation of∆ sl ( T , )∆ sl ( ) = t t + t + t + t t − + t − t + t − t − . (4)Note that Theorem 1.2 could also be stated in terms of ∆ sl and ∆ sl , as ∆ sl on knots equalsthe Alexander polynomial evaluated at t . This motivates the following conjecture. For the MATTHEW HARPER reasons outlined above regarding the difficulty in proving Theorem 1.2, this conjecture is notobvious and would require a more efficient method to prove it in general.
Conjecture 1.3.
Let k be the Lie algebra obtained by removing m simple roots α j of a simply-laced Lie algebra g of rank n . Suppose the corresponding entries t j of t ∈ ( C × ) n are chosen sothat the representation V ( t ) of U ζ ( g ) has m maximal subrepresentations. Let ˆ t ∈ ( C × ) n − m beobtained by removing the entries t j from t . Then for any knot K , ∆ g ( K )( t ) = ∆ k ( K )( ˆ t m +1 ) . The sl invariant admits a nine-term skein relation via the characteristic polynomial of the R -matrix represented in V ( t ) ⊗ V ( t ), see Proposition 6.5. Using a recursion determined bythe square of the R -matrix, we have computed an explicit formula for (2 n + 1 ,
2) torus knots.
Theorem 1.4 (Two Strand Torus Knots) . The value of ∆ sl on a (2 n + 1 , torus knot isgiven by: ( t − t − )( t n +21 + t − (4 n +2)1 )( t + t − )( t + t − )( t t − t − t − ) + ( t − t − )( t n +22 + t − (4 n +2)2 )( t + t − )( t + t − )( t t − t − t − )+ ( t t + t − t − )( t n +21 t n +22 + t − (4 n +2)1 t − (4 n +2)2 )( t t + t − t − )( t + t − )( t + t − ) . The following representation-theoretic discussion assumes g = sl . Let P ∼ = ( C × ) denotethe characters on the Cartan subalgebra. Note that P has a group structure under entrywisemultiplication with identity = (1 , Lemma 1.5 ([Har19b]) . The representation V ( t ) is reducible if and only if t belongs to anyof X i = { t ∈ P : t i = 1 } or X = { t ∈ P : ( t t ) = − } . (5)Note that the X α are indexed by the set of positive roots Φ + . Let R = (cid:83) α ∈ Φ + X α and R α = X α \ (cid:16)(cid:83) β ∈ Φ + \{ α } X β (cid:17) . Definition 1.6.
Let X ( t ) , Y ( t ) , and W ( t ) denote the head of V ( t ) for t belonging to exactlyone of X , X , or X , respectively.Let ν = ( − , ζ ) and ν = ( ζ, −
1) be elements of P . If t ∈ R , then V ( t ) is reducible and itfits into at least one exact sequence below:0 → X ( ν t ) → V ( t ) → X ( t ) → → Y ( ν t ) → V ( t ) → Y ( t ) → → W ( ν ν t ) → V ( t ) → W ( t ) → . (8)Note that if t belongs to two of the defining sets of R , the corresponding quotients of V ( t )are reducible and V ( t ) belongs to two of the above sequences. Conversely, if V ( t ) belongs totwo of the above sequences then t belongs to some pairwise intersection of X , X and X .In this case, V ( t ) has four, rather than two, composition factors in its Jordan-H¨older series. Theorem 1.7 (Constructions of the Alexander Polynomial) . The invariant of a link whosecomponents are colored by a representation X ( t ) , Y ( t ) , or W ( t ) is the Alexander-Conwaypolynomial evaluated at t . GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 5
Like Theorem 1.2, this theorem cannot be proven directly from the skein relation determinedby the characteristic polynomial of the R -matrix. Rather, we verify the Conway relationagainst an arbitrary tangle under the modified trace. A key step in showing this makes useof the following tensor product decompositions. Theorem 1.8 (Tensor Square Decompositions) . For each isomorphism below, we assume t , s ∈ P are chosen so that the four-dimensional representations which appear are well-defined and all summands are irreducible: X ( t ) ⊗ X ( s ) ∼ = X ( ts ) ⊕ X ( ν ν ts ) ⊕ V ( ν ts ) (9) Y ( t ) ⊗ Y ( s ) ∼ = Y ( ts ) ⊕ Y ( ν ν ts ) ⊕ V ( ν ts ) (10) W ( t ) ⊗ W ( s ) ∼ = W ( ν ts ) ⊕ W ( ν ts ) ⊕ V ( ts ) . (11)1.2. Relation to Other Invariants.
There are recent discoveries, similar in flavor to thecurrent work, relating invariants from the quantum supergroups gl m | n and the Alexanderpolynomial. The Links-Gould invariants LG m,n are conjectured to satisfy the relation LG m,n ( L )( t, e iπ/m ) = (∆( L )( t m )) n (12)for all ( m, n ). This conjecture was proven for all ( m,
1) in [DIL05], and for all (1 , n ) in[KP17]. Compare this with Conjecture 1.3 above.Our observation that ∆ sl assigns non-trivial polynomials to knots with trivial Alexandermodules implies it is a non-abelian invariant in the sense of Cochran, but unlike Cochran’s A Z n invariants, it distinguishes knots with trivial Alexander polynomial [Coc04]. Neverthe-less, we suspect ∆ sl is related to other geometrically constructed invariants that are sensitiveto knots with trivial Alexander modules. Knot Floer homology, for example, is non-trivialon the Whitehead double of [Hed07]. Another example is the set of twisted Alexanderpolynomials for a particular matrix group [Wad94]. The set of twisted invariants derivedfrom all parabolic SL ( F ) representations, up to conjugacy, of the knot groups of n34 and n42 are enough to distinguish the pair of mutant knots from each other and the unknot.Another attempt at refining Alexander invariants by passing to higher-rank Lie types arethe SU ( n ) Casson invariants, developed by Frohman [Fro93]. However, these invariants forfibered knots are completely determined by their Alexander polynomials [BN00]. Since and are fibered, their SU ( n ) Casson invariants are identical. These knots are distin-guished by ∆ sl , demonstrating it is a stronger invariant on fibered knots.It is also shown in [BCGP16] that the Reidemeister torsion is recovered from TQFTs basedon the sl representations V ( t ). We expect that applying their TQFT to higher rank quan-tum groups at a fourth root of unity generalizes Reidemeister torsion and implies a Turaevsurgery formula [Tur02] in terms of ∆ g .In rank one, Ohtsuki exhibits an isomorphism between the braid group representation de-termined by tensor powers of V ( t ) and exterior powers of the Burau representation [Oht02].The proof relies on the tensor decomposition formula of V ( t ) ⊗ V ( t ) and uses the basisvectors of this decomposition to compute partial traces of intertwiners. This identificationrecovers the determinant formula for the Alexander polynomial. Further investigation ofthe braid representations from X ( t ), Y ( t ), and W ( t ) may uncover a higher rank geometric MATTHEW HARPER construction of the Burau representation. This geometric interpretation could then extendto ∆ sl .1.3. Structure of Paper.
In Section 2 we recall the restricted quantum group U ζ ( g ). Wealso define a bilinear pairing on the subalgebra generated by F and F then prove it isnon-degenerate.We recall the representations V ( t ) in Section 3. The pairing defined in the previous sectionallows us to prove a duality property for V ( t ) in types A, D, E, G, and B even . Specializingto sl and t ∈ R α , we prove various results on the irreducible representations which appearin the composition series for V ( t ). Their relation to the Alexander polynomial is proven inSection 5.Section 4 introduces the unrolled structure of U ζ ( g ) and recalls the formula for the R -matrix.For g = sl , we give the explicit R -matrix action on the V ( t ) ⊗ V ( t ) expressed in terms ofthe direct sum decomposition computed in [Har19b].We give an overview on computing invariants in Section 5, then prove that the four-dimensionalrepresentations X ( t ), Y ( t ), and W ( t ) yield the Alexander polynomial in the variable t forany link L .Section 6 is concerned with the properties of ∆ g . In this section, we prove the symmetriesenjoyed by ∆ sl , discuss its skein relation, give a method to compute ∆ sl for families of torusknots, and prove Theorem 1.2.The values of ∆ sl and several observations are presented in Section 7.1.4. Acknowledgments.
I am very grateful to Thomas Kerler for insightful discussions. Iwould also like to extend my thanks to Sachin Gautam for helpful comments. I thank theNSF for support through the grant
NSF-RTG during Spring 2020.Results in Sections 4, 5, 6, and 7 rely on computations done in Maple 2018.0. The code canbe downloaded from github.com/harperrmatthew/sl3invariant .2.
Restricted Quantum Groups
In this section, we recall the restricted quantum group U ζ ( g ) at a primitive fourth root ofunity via generators and relations, as given in [Har19b]. We define a bilinear pairing onnegative root vectors and prove in Corollary 2.5 that this pairing is non-degenerate. We willmake use of this pairing in the next section to state a duality property on induced repre-sentations. Throughout this section, we fix a primitive fourth root of unity ζ and g is anysemisimple Lie algebra, unless noted otherwise.Let A denote the Cartan matrix of g , which is symmetrized by d . As described in [CP95],we fix an ordering < br on Φ + according to the braid actions T i determined by a presentationof the longest word of the Weyl group. These actions define the non-simple root vectors E α and F α as elements of Lusztig’s divided powers algebra U divζ ( g ) [Lus90b]. Suppose E α in U divζ ( g ) does not belong to the subalgebra generated by simple root vectors over Z [ ζ ], or GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 7 that α belongs to the Weyl group orbit of α i and d i = 2. Then the root α is said to be negligible . Let Φ denote the set of negligible root vectors. Then Φ + = Φ + \ Φ denotes thepositive roots of g which are not negligible. Define ∆ + to be the simple roots in Φ + . Fora root α = (cid:80) c j ( α ) α j , we define g ( α ) to be the greatest index i for which c i ( α ) (cid:54) = 0. Let h ( α ) = (cid:80) c j ( α ) be the height of α and h (cid:48) ( α ) = c i ( α ) − h ( α ). Let (cid:98) x (cid:99) d = x − x − ζ d − ζ − d , (13)omitting subscripts when d = 1. The following definition is a consequence of Lusztig [Lus90a,Lus90b]. Definition 2.1.
The restricted quantum group U ζ ( g ) is the Q ( ζ )-algebra generated by E α , F α , for α ∈ Φ + , and K j for 1 ≤ j ≤ n with relations: K i K − i = 1 , K i K j = K j K i , (14) K i E α j = q d i A ij E α j K i , K i F α j = q − d i A ij F α j K i , (15)[ E α i , F α j ] = δ ij (cid:98) K i (cid:99) d i , E α = F α = 0 , (16) E α E α i = E α i E α F α F α i = F α i F α (cid:27) for ( α, α i ) = 0 , i < g ( α ) , and h (cid:48) ( α ) ∈ Z , (17) E β E α = ζE α E β + ζE α + β E β E α + β = − ζE α + β E β E α E α + β = ζE α + β E α F α F β = − ζF β F α − ζF α + β F β F α + β = − ζF α + β F β F α F α + β = ζF α + β F α for ( α, β ) = − β = α i and i < g ( α ) or h ( β ) = h ( α ) + 1 and g ( α ) = g ( β ) . (18)In the sl case, we take non-simple root vectors E = − ( E E + ζE E ) and F = − ( F F − ζF F ) , (19)and order Φ + according to α < br α + α < br α . (20)We may use α to mean the non-simple root α + α . For brevity, we will denote U ζ ( g ) by U when no confusion arises. There is Hopf algebra structure on U , but we will only needthe description of the antipode here: S ( E i ) = − E i K − i S ( F i ) = − K i F i S ( K i ) = K − i . (21)Our other conventions can be found in [Har19b]. Let Ψ denote the space of maps { , } Φ + . Theorem 2.2 ([Lus90b]) . The restricted quantum group U has a PBW basis { E ψ F ψ (cid:48) K k : ψ, ψ (cid:48) ∈ Ψ and k ∈ Z n } , (22) where E ψ = (cid:81) α ∈ Φ + E ψ α α , F ψ = (cid:81) α ∈ Φ + F ψ (cid:48) α α , K k = (cid:81) ni =1 K k i i , and the products are orderedwith respect to < br . MATTHEW HARPER
Upon deletion of negligible simple roots, it is straightforward to verify the following quantumgroup isomorphisms: U ζ ( b n ) ∼ = U ζ ( a n − )[ K ± n ] / (cid:104){ K n , E n − } , { K n , F n − }(cid:105) (23) U ζ ( c n ) ∼ = U ζ ( a )[ K ± , . . . , K ± n ] / (cid:104){ K , E } , { K , E }(cid:105) (24) U ζ ( f ) ∼ = U ζ ( a )[ K ± , K ± ] / (cid:104){ K , E } , { K , F }(cid:105) , (25)and that the subalgebras generated by negative root vectors of each of U ζ ( g ) and U ζ ( a )are isomorphic.We define the subalgebras U = (cid:104) K ± i : 1 ≤ i ≤ n (cid:105) , U + = (cid:104) E α : α ∈ ∆ + (cid:105) , and U − = (cid:104) F α : α ∈ ∆ + (cid:105) . (26) Lemma 2.3 ([Har19b]) . Let ψ ∈ Ψ and ψ (cid:48) = 1 − ψ. Then F ψ F ψ (cid:48) is a nonzero multiple of F (1 ... . For each ψ ∈ Ψ, we define χ ψ : U − → Q ( ζ ) so that χ ψ ( F ) is the coefficient of F ψ in thePBW basis expression for any F ∈ U − . Lemma 2.4 ([Har19b]) . Let ψ , ψ ∈ Ψ such that ψ < ψ . Then χ (1 ... ( F − ψ F ψ ) = 0 . Observe that each χ ψ determines a bilinear pairing on U − . For each F, F (cid:48) ∈ U − , we define( F, F (cid:48) ) ψ = χ ψ ( F F (cid:48) ). Corollary 2.5.
The bilinear pairing ( · , · ) (1 ... is non-degenerate.Proof. Suppose X := (cid:80) a ψ F ψ ∈ U − is degenerate with respect to ( · , · ) (1 ... , i.e. (cid:104) F, X (cid:105) = 0for all F ∈ U − . Let a ψ ∗ be the nonzero coefficient of greatest index in X . By Lemma 2.3,( F − ψ ∗ , F ψ ∗ ) (1 ... (cid:54) = 0. Using Lemma 2.4, for each ψ < ψ ∗ , ( F − ψ ∗ , F ψ ) (1 ... = 0. Thus,( F − ψ ∗ , X ) (1 ... = a ψ ∗ ( F − ψ ∗ , F ψ ∗ ) (1 ... = 0 . (27)This contradicts that a ϕ is nonzero. Thus, X = 0, which proves non-degeneracy of ( · , · ) (1 ... . (cid:3) Representations of U ζ ( g )Here we recall the representation V ( t ) as a Verma module over U . In Proposition 3.4, we usethe non-degenerate pairing defined in the previous section to show that V ( t ) ∗ is isomorphicto V ( − t − ) in types A, D, E, G, and B even . We then specialize to U ζ ( sl ). We characterizethe structure of V ( t ) when it has a four-dimensional irreducible subrepresentation. Thesecases are studied in Propositions 3.8 and 3.10. In Theorems 1.8 and 3.11, we state the tensorproduct decompositions for these representations.Let P denote the characters on (cid:104) K ± , . . . , K ± n (cid:105) . Note that P has a group structure underentrywise multiplication with identity = (1 , . . . , P ∼ = ( C × ) n under theidentification of each t ∈ P with its values on K i . Let B = (cid:104) E α , K ± i : α ∈ ∆ + , ≤ i ≤ n (cid:105) bethe Borel subalgebra. Each character t ∈ P extends to a character γ t : B → C by γ t ( K i ) = t i and γ t ( E i ) = 0 . (28) GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 9
Definition 3.1.
Let γ t : B → C be a character as in (28). Let V t = (cid:104) v (cid:105) be theone-dimensional left B -module determined by γ t , i.e. for each b ∈ B , bv = γ t ( b ) v . Wedefine the representation V ( t ) to be the induced module V ( t ) = Ind UB ( V t ) = U ⊗ B V t . (29)By Theorem 2.2, we have that V ( t ) ∼ = U − as vector spaces. Lemma 3.2.
Suppose g is of type A, D, E, G, or B even . Then the lowest weight of V ( t ) is − t .Proof. First, suppose that g is of ADE type. As F (1 ... is the product of all positive rootvectors, K i F (1 ... = ζ ( Aρ ) i F (1 ... K i , with the vector ρ denoting the sum of positive root vec-tors expressed in the basis of simple roots α j . We refer to [Bou02] for the Cartan and rootdata. It is a routine computation to verify that ζ ( Aρ ) i = − i and g of ADE type.We now assume that g is a Lie algebra of type B n . Then U ζ ( g ) ∼ = U ζ ( a n − )[ K n ] / (cid:104){ K n , E n − } , { K n , F n − }(cid:105) .Let ρ be the sum of non-negligible roots expressed in the basis of simple roots α j . We checkthat ζ ( dAρ ) n = −
1. Since α n − appears ( n − n − ( n −
1) + 1) = 2( n −
1) times in ρ = (cid:80) n − i =1 i ( n − i + 1) α i , commuting K n past (cid:81) α ∈ Φ + F α yields a factor ζ − n − = ( − n − .Thus, the claim holds for even n .As noted above, the subalgebras generated by negative root vectors in restricted quantumgroups of type G and A are isomorphic. In type G , we have K i F (1 ... = ( − ζ ) ( Aρ ) i F (1 ... K i ,with A denoting the Cartan matrix of type A . Thus, the lemma holds for type G . (cid:3) Remark 3.3.
The existence of central Cartan generators in types F and C n for n > Proposition 3.4.
Let g be of type A, D, E, G, or B even . Then for all t ∈ P , V ( t ) ∗ ∼ = V ( − t − ) .Proof. We consider all types simultaneously. The lowest weight vector of V ( t ) is F (1 ... v = (cid:81) α ∈ Φ + F α v . For each ψ ∈ Ψ, let f ψ be the dual vector which evaluates to one on F ψ v andis zero otherwise. Observe that f (1 ... is a highest weight vector for V ( t ) ∗ . Indeed, for any v ∈ V ( t ), E i · f (1 ... ( v ) = f (1 ... ( − E i K − i v ) = 0 , (30)since − E i K − i v cannot be lowest weight. We claim that { F ψ · f (1 ... : ψ ∈ Ψ } is a basisfor V ( t ) ∗ . Let (cid:104)· , ·(cid:105) be the bilinear pairing defined as (cid:104) F, F (cid:48) (cid:105) = F · f (1 ... ( F (cid:48) v ) for every F, F (cid:48) ∈ U − . It follows that F ψ · f (1 ... = (cid:80) ψ (cid:48) ∈ Ψ (cid:104) F ψ , F ψ (cid:48) (cid:105) f ψ (cid:48) and { F ψ · f (1 ... : ψ ∈ Ψ } isa basis if and only if (cid:104)· , ·(cid:105) is non-degenerate. Recall the pairing ( F, F (cid:48) ) (1 ... = χ (1 ... ( F F (cid:48) )defined in Section 2. We have for each ψ, ψ (cid:48) ∈ Ψ, (cid:104) S − ( F ψ ) , F ψ (cid:48) (cid:105) = f (1 ... ( F ψ F ψ (cid:48) v ) = ( F ψ , F ψ (cid:48) ) (1 ... . (31)Since ( · , · ) (1 ... is non-degenerate, the dual vectors S − ( F ψ ) · f (1 ... form a basis of V ( t ) ∗ .Since S and each K i is invertible, S determines a change of basis on V ( t ) ∗ . Therefore, thedual vectors F ψ · f (1 ... form a basis of V ( t ) ∗ . Thus, V ( t ) ∗ ∼ = V ( s ) for some s ∈ P . Since K i acts by K − i on the dual representation, s is inverse to the weight of F (1 ... v . By Lemma 3.2,it follows that s = − t − . Thus, V ( t ) ∗ ∼ = V ( − t − ). (cid:3) For the remainder of this section, we suppose g = sl . By Theorem 2.2, B = { , F , F , F F , F , F F , F F , F F F } (32)= { F (000) , F (100) , F (001) , F (101) , F (010) , F (110) , F (011) , F (111) } (33)is an ordered basis of U − . Moreover, B determines the standard basis of V ( t ) by tensoringbasis vectors of U − with v .We give the actions of E and E on the standard basis in Table 1 below. We also providea graphical description of the action of U on V ( t ) in terms of weight spaces labeled bythe standard basis in Figure 3. Each solid vertex indicates a one-dimensional weight spaceof V ( t ), and the “dotted” vertex indicates the two-dimensional weight space spanned by F (101) v and F (010) v . An edge is drawn between vertices if the action of either E or E isnonzero between the associated weight spaces. We do not assign edges to matrix elements of F and F , since they are independent of t . However, for non-generic choices of the parameter t , edges are deleted from the graph because matrix elements of E and E vanish. We orientthe graph so that F acts downward left and F acts downward right at each vertex, andeach E i acts in the opposite direction of the corresponding F i . Table 1.
Actions of E and E on V ( t ) expressed in the standard basis. Theremaining actions are zero for all t ∈ P . E F (100) v = (cid:98) t (cid:99) F (000) v E F (001) v = (cid:98) t (cid:99) F (000) v E F (101) v = (cid:98) ζt (cid:99) F (001) v E F (101) v = (cid:98) t (cid:99) F (100) v E F (010) v = ζt F (001) v E F (010) v = − t − F (100) v E F (101) v = ζt F (101) v − (cid:98) ζt (cid:99) F (010) v E F (011) v = t − F (101) v + (cid:98) t (cid:99) F (010) v E F (111) v = (cid:98) t (cid:99) F (011) v E F (111) v = (cid:98) t (cid:99) F (110) v F (111) v F (110) v F (011) v F (101) v , F (010) v F (100) v F (001) v v Figure 3.
The action of U on the weight spaces of V ( t ).We now state the genericity condition on V ( t ). Let X = { t ∈ P : t = 1 } , X = { t ∈ P : t = 1 } X = { t ∈ P : ( t t ) = − } , (34) GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 11 then set R to be the union of X , X , and X . We partition R into disjoint subsets indexedby nonempty subsets I (cid:40) Φ + , with R I = (cid:32) (cid:92) α ∈ I X α (cid:33) \ (cid:32) (cid:91) α/ ∈ I X α (cid:33) (35) Proposition 3.5 ([Har19b]) . The representation V ( t ) of U is irreducible if and only if t / ∈ R . If t belongs to R , R , or R then the socle of V ( t ) is an irreducible maximal subrepre-sentation of dimension four. Moreover, the head is four-dimensional, irreducible, and hashighest weight t . We use B i to denote the subalgebra of U generated by B and F i . Definition 3.6.
Suppose t ∈ X . Let γ X t be the extension of the character γ t on B to B with γ X t ( F ) = 0. Set X t = (cid:104) x (cid:105) to be the one-dimensional B -module determined by γ X t and define X ( t ) = Ind UB ( X t ) = U ⊗ B X t . (36)The representation Y ( t ) is defined analogously by assuming t ∈ X and setting the actionof F to be zero on the generating vector. Remark 3.7. If t / ∈ X , then X ( t ) is not defined. Observe that 0 = [ E , F ] x = (cid:98) K (cid:99) x = (cid:98) t (cid:99) x is only satisfied if t = 1.Let ν = ( − , ζ ) and ν = ( ζ, −
1) belong to P . These characters correspond to the multi-plicative weight shifts under the actions of F and F , respectively. Proposition 3.8. If t ∈ X or if t ∈ X , we have the respective exact sequences → X ( ν t ) → V ( t ) → X ( t ) → → Y ( ν t ) → V ( t ) → Y ( t ) → . (38)As a subrepresentation of V ( t ), X ( ν t ) has a basis given by (cid:104) F (001) v , F (101) v , F (011) v , F (111) v (cid:105) .It can be seen as a subrepresentation in Figure 4, given below. The figure is obtained fromFigure 3 by deleting the edges where both E and E act by zero under the given assumptionson t . Moreover, assuming t ∈ R is equivalent to assuming both X ( ν t ) and its quotient in V ( t ) are irreducible. Analogous statements are true for Y ( ν t ).Let B = (cid:104) B, E F F , E F F (cid:105) and γ W t be the character on B which equals γ t on B andevaluates to zero otherwise. Definition 3.9.
Let t ∈ X and let W t = (cid:104) w (cid:105) be the one-dimensional B -module deter-mined by γ W t . We define W ( t ) by induction W ( t ) = Ind UB ( W t ) = U ⊗ B W t . (39)Given that t ∈ X , the vectors F F w and F w are proportional in W ( t ) and implies W ( t ) is four-dimensional. Note that W ( ν ν t ) can be considered a subrepresentation of V ( t ) with basis (cid:104) E E F (111) v , E F (111) v , E F (111) v , F (111) v (cid:105) . Proposition 3.10. If t ∈ X , we have the following exact sequence: → W ( ν ν t ) → V ( t ) → W ( t ) → . (40) F (111) v F (110) v F (011) v F (101) v , F (010) v F (100) v F (001) v v t ∈ R F (111) v F (110) v F (011) v F (101) v , F (010) v F (100) v F (001) v v t ∈ R Figure 4.
Reducible V ( t ) with subrepresentations X ( ν t ) and Y ( ν t ).In Figure 5, we assume t ∈ R so that both W ( t ) and W ( ν ν t ) are irreducible. To distin-guish this figure from Figure 3 we draw a disconnected graph. This should not be interpretedas a direct sum decomposition, as V ( t ) is indecomposable for all t ∈ P . F (101) v , F (010) v F (001) v F (100) v v F (111) v F (011) v F (110) v t ∈ R Figure 5.
Reducible V ( t ) with subrepresentation W ( ν ν t ).We give the decomposition rules for tensors between the same representation types. Theorem 1.8 (Tensor Square Decompositions) . For each isomorphism below, we assume t , s ∈ P are chosen so that the four-dimensional representations which appear are well-defined and all summands are irreducible: X ( t ) ⊗ X ( s ) ∼ = X ( ts ) ⊕ X ( ν ν ts ) ⊕ V ( ν ts ) (9) Y ( t ) ⊗ Y ( s ) ∼ = Y ( ts ) ⊕ Y ( ν ν ts ) ⊕ V ( ν ts ) (10) W ( t ) ⊗ W ( s ) ∼ = W ( ν ts ) ⊕ W ( ν ts ) ⊕ V ( ts ) . (11) Proof.
We prove the isomorphism in (9) by considering the map which sends the highestweight vectors of V ( ts ), V ( ν ν ts ), and V ( ν ts ) to the respective highest weight vectorsin X ( t ) ⊗ X ( s ): x ⊗ x , ∆( E E E ) . ( F F F x ⊗ F F F x ) , and ∆( E ) . ( F x ⊗ F x ) . (41) GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 13
After simplifying the expressions for these vectors, one finds that they are nonzero and havedistinct weights, given our assumptions on t and s . Note that V ( ts ) and V ( ν ν ts ) are re-ducible, but their heads are irreducible and have highest weights ts and ν ν ts respectively.We also have that V ( ν ts ) is irreducible by assumption. Thus, the head of each of V ( ts ), V ( ν ν ts ), and V ( ν ts ) is sent to a distinct nonzero subspace of X ( t ) ⊗ X ( s ). The soclesof V ( ts ) and V ( ν ν ts ) are irreducible and have highest weights ν ts and − ts . Therefore,they must be sent to zero. By dimensionality, this map is an isomorphism. Similarly for (10).The same argument also holds for W ( t ) ⊗ W ( s ), but in this case the respective generatingvectors are∆( E ) . ( F w ⊗ F w ) , ∆( E ) . ( F w ⊗ F w ) and w ⊗ w . (42)Here we have used w to denote the highest weight vector of W ( t ). (cid:3) Although we will not use them in this paper, we include the data of mixed tensor productsfor completeness.
Theorem 3.11 (Mixed Tensor Decomposition) . For each isomorphism below, we assume t , s ∈ P are chosen so that the four-dimensional representations which appear are well-defined and all summands are irreducible: X ( t ) ⊗ Y ( s ) ∼ = V ( ts ) ⊕ V ( ν ν ts ) (43) X ( t ) ⊗ W ( s ) ∼ = V ( ts ) ⊕ V ( ν ts ) (44) Y ( t ) ⊗ W ( s ) ∼ = V ( ts ) ⊕ V ( ν ts ) (45) V ( t ) ⊗ X ( s ) ∼ = V ( ts ) ⊕ V ( ν ts ) ⊕ V ( ν ν ts ) ⊕ V ( ν ν ts ) (46) V ( t ) ⊗ Y ( s ) ∼ = V ( ts ) ⊕ V ( ν ts ) ⊕ V ( ν ν ts ) ⊕ V ( ν ν ts ) (47) V ( t ) ⊗ W ( s ) ∼ = V ( ts ) ⊕ V ( ν ts ) ⊕ V ( ν ts ) ⊕ V ( ν ν ts ) . (48) Proof.
Using the same argument as above, we only provide highest weight vectors whichgenerate an irreducible representation under the action of F and F . We then check theweights of these generating vectors, which indicate the isomorphism class of the resultingrepresentation. We omit cases below involving Y ( t ) when the result can be determined fromthe expression for X ( t ) by switching the indices 1 and 2. X ( t ) ⊗ Y ( s ) : x ⊗ y , ∆( E E E E ) . ( F F F x ⊗ F F F y ) X ( t ) ⊗ W ( s ) : x ⊗ w , ∆( E ) . ( F x ⊗ F w ) V ( t ) ⊗ X ( s ) : v ⊗ x , ∆( E )( F v ⊗ F x ) , ∆( E E E E )( F F F v ⊗ F F F x ) , ∆( E E E E )( F F F v ⊗ F F F F x ) V ( t ) ⊗ W ( s ) : v ⊗ w , ∆( E ) . ( F v ⊗ F w ) , ∆( E ) . ( F v ⊗ F w ) , ∆( E E E E ) . ( F F v ⊗ F F F F w ) (cid:3) Unrolled Restricted Quantum Groups and Braiding
We now consider the unrolled restricted quantum group. In this context, we may definean R -matrix using the action of H , . . . , H n . Therefore, the category of unrolled restrictedquantum groups is braided. We end this section with the R -matrix action for quantum sl on the tensor decomposition of V ( t ) ⊗ V ( t ). Definition 4.1.
Let g be a semisimple Lie algebra of rank n . We define unrolled restrictedquantum g at a fourth root of unity U H to be the algebra U [ H , . . . , H n ] with relations: H i K ± j = K ± j H i , [ H i , E j ] = A ij E j , [ H i , F j ] = − A ij F j (49)in addition to the relations of U itself.The representation V ( t ) extends from the restricted quantum group to the unrolled re-stricted quantum group. Fix a character t ∈ P and suppose ζ = e πi/ . Choose λ i suchthat ζ λ i = e πiλ i / = t i . Since t i (cid:54) = 0 there are infinitely many choices of λ i . By an abuseof notation, we will suppose the unrolled restricted quantum group acts on V ( t ) so that H i v = λ i v for some fixed λ i satisfying the condition above. However, the choice of λ i willnot be relevant for our purposes.We consider any representation for which H i v = λ i v . Once given an action of H i , wecan define the R -matrix action on tensor product representations. The R -matrix can benormalized to depend only on ζ λ and not λ itself. The formula for the universal R -matrix isgiven explicitly in [CP95, Theorem 8.3.9] for simple g and is truncated under the assumption q = ζ . For weight representations V and W , we define the map R : V ⊗ W → V ⊗ W by R = ζ ( (cid:80) ij ( A − ) ij H i ⊗ H j ) (cid:89) α ∈ Φ + (1 ⊗ ζ − ζ − ) E α ⊗ F α ) , (50)with the action of ζ ( (cid:80) ij ( A − ) ij H i ⊗ H j ) on weight vectors v and w given by ζ ( (cid:80) ij ( A − ) ij H i ⊗ H j )( v ⊗ w ) = ζ ( (cid:80) ij ( A − ) ij µ i ν j )( v ⊗ w ) , (51)with H i ⊗ H j ( v ⊗ w ) = µ i ν j ( v ⊗ w ), and the product over Φ + follows the ordering < br . Notethat the expression ζ ( (cid:80) ij ( A − ) ij H i ⊗ H j ) does not belong to U H ⊗ U H , as it is defined in termsof power series in H i . Let P be the map that swaps tensor factors. Then the category ofweight representations of U H has a braiding given by P ◦ R .We define R t to be the action of P ◦ R on the representation V ( t ) ⊗ V ( t ) normalized by theribbon element so that each partial quantum trace of R t is the identity. Since the ribbonelement is central, it acts by a scalar called the writhe factor . Under this normalization, ∆ g is an invariant of unframed links.Let σ ψ denote the weight of F ψ v h in V ( ). Ordered according to the standard basis, the σ ψ in type A are:(1 , , ( − , ζ ) , ( ζ, − , ( − ζ, − ζ ) , ( − ζ, − ζ ) , ( ζ, , (1 , ζ ) , ( − , − . (52)A pair of characters ( t , s ) ∈ P is called non-degenerate if V ( σ ψ ts ) is irreducible for each ψ ∈ Ψ. Theorem 4.2 ([Har19b]) . Let ( t , s ) be a non-degenerate pair. The tensor product V ( t ) ⊗ V ( s ) decomposes as a direct sum of irreducibles according to the formula V ( t ) ⊗ V ( s ) ∼ = (cid:77) ψ ∈ Ψ V ( σ ψ ts ) . (53) GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 15
Corollary 4.3.
Suppose that g = sl and that ( t , t ) ∈ P is non-degenerate. Under thetensor product decomposition of V ( t ) ⊗ V ( t ) , we have V ( t ) ⊗ V ( t ) V ( t ) ⊗ V ( t ) (cid:76) ψ ∈ Ψ V ( σ ψ t ) (cid:76) ψ ∈ Ψ V ( σ ψ t ) R t ∼ = ∼ = r ⊗ id × , (54) with r given by diag ( t t , − t , − t ) ⊕ (cid:2) − ξ − ξ (cid:3) ⊕ diag ( − t − , − t − , t − t − ) (55) in the basis of summands V ( σ ψ t ) ordered according to the standard basis of V ( ) .Proof. Since ( t , t ) is a non-degenerate pair, each summand is irreducible. Therefore, R t actsby a constant on each multiplicity-one summand and acts by an amplified 2 × r . (cid:3) Invariants from Unrolled Restricted Quantum sl The goal of this section is to prove Theorem 1.7. We begin by recalling the steps used in[Oht02], with a different convention, to obtain an unframed invariant of oriented 1-tangles(or long knots) from representations of a ribbon Hopf algebra. Because the quantum dimen-sion of these representations we are considering is zero, we must compute the invariant viaa modified trace to obtain a meaningful result.Let (
V, ρ ) be a U H -module over a field K such that each H i acts diagonally. To assign linearmaps to tangles, we use the convention that an upward pointing strand is the identity on V ,and a downward pointing strand is the identity on V ∗ . Recall evaluation and coevaluation ontheir tensor products, which allow us to define a partial quantum trace on representations.Figure 6 exhibits the duality maps on V associated to oriented “cups” and “caps”. Thesemaps satisfy the relations( id V ⊗ ev )( coev ⊗ id V ) = id V = ( (cid:101) ev ⊗ id V )( id V ⊗ (cid:103) coev ) (56)and ( ev ⊗ id V ∗ )( id V ∗ ⊗ coev ) = id V ∗ = ( id V ∗ ⊗ (cid:101) ev )( (cid:103) coev ⊗ id V ∗ ) . (57) ∼ ev ∈ Hom( V ∗ ⊗ V, K ) ∼ (cid:101) ev ∈ Hom( V ⊗ V ∗ , K ) ∼ coev ∈ Hom( K , V ⊗ V ∗ ) ∼ (cid:103) coev ∈ Hom( K , V ∗ ⊗ V ) Figure 6.
Conventions for cups and caps.Let (cid:80) r i α i denote the sum of all positive roots of g . Let h V = ρ ( (cid:81) K r i i ) be the pivotalelement, which we use to define quantum trace. Given any basis ( e i ) of V and a corresponding dual basis ( e ∗ i ), the above maps are defined as ev ( e ∗ i ⊗ e j ) = e ∗ i ( e j ) , (cid:101) ev ( e i ⊗ e ∗ j ) = e ∗ j ( h V e i ) , (58) coev (1) = (cid:88) i e i ⊗ e ∗ i , (cid:103) coev (1) = (cid:88) i e ∗ i ⊗ h − V e i , (59)and do not depend on the choice of basis. Remark 5.1.
Let tr n denote partial trace over the n -th tensor factor. For an intertwiner A : V ⊗ n → V ⊗ n , the n -th partial quantum trace of A tr n (cid:0) ( id ⊗ n − ⊗ h V ) A (cid:1) : V ⊗ n − → V ⊗ n − (60)is also an intertwiner. Assuming V is irreducible, tr ...n (cid:0) ( id ⊗ h ⊗ n − V ) A (cid:1) acts by a scalar a on V . Since tr( h V ) = 0, it follows thattr( h ⊗ nV A ) = tr (cid:0) tr ...n ( h ⊗ nV A ) (cid:1) = a tr( h V ) = 0 (61)and tr (cid:0) ( id ⊗ h ⊗ n − V ) A (cid:1) = tr (cid:0) tr ...n (cid:0) ( id ⊗ h ⊗ n − V ) A (cid:1)(cid:1) = a tr( id V ) = a dim( V ) . (62) Definition 5.2.
Let b ∈ B n and V a U H -module. We define ψ n ( b ) to be the action of b on V ⊗ n , where each braid group generator σ i is identified with the action of P ◦ R in the i th and i + 1 st tensor positions of V ⊗ n . We assign to b the quantity∆ V ( b ) = 1dim V tr (cid:0) ( id V ⊗ h ⊗ n − V ) ψ n ( b ) (cid:1) . (63)It follows from [Oht02] that ∆ V ( b ) is an invariant of the braid closure. In particular, it is aninvariant of links. If (cid:98) b = L , we will denote the invariant of L by ∆ V ( L ) = ∆ V ( b ).For V = V ( t ), we define ∆ g ( L ) to be the Laurent polynomial in n = rank( g ) variables givenby ∆ V ( t ) ( L ). We now consider ∆ V for V some four-dimensional irreducible representationof U Hζ ( sl ). Theorem 1.7 (Constructions of the Alexander Polynomial) . The invariant of a link whosecomponents are colored by a representation X ( t ) , Y ( t ) , or W ( t ) is the Alexander-Conwaypolynomial evaluated at t .Proof. Fix a four-dimensional irreducible representation (
V, ρ ). Let R V denote the restric-tion of R t to V . By the above, ∆ V is an unframed link invariant, which evaluates to 1 onthe unknot. It is sufficient to show that it satisfies the Alexander-Conway skein relation inthe variable ( t ) . = ( t − t − ) − Figure 7.
Alexander-Conway skein relation in the variable ( t ) GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 17
However, this relation does not follow immediately from the R -matrix, as it does in rankone. Set δ V = R V − R − V − ( t − t − ) id V ⊗ . (64)Given any link L , we select a region on which we will apply the relation and comb the linkinto a closed braid presentation so that it is at the top left of the braid diagram. “Cutting”the two left-most strands of L in this presentation, we obtain an intertwiner on V ⊗ , whichwe denote by A . We claim that tr (cid:0) ( id V ⊗ h V ) δ V A (cid:1) = 0 . (65)To show this, let Γ V be the map which sends V ⊗ to its direct sum decomposition determinedby Theorem 1.8. After explicit computation, we findΓ − V δ V Γ V = d
00 0 0 d ⊗ id × =: (cid:99) δ V ⊗ id × (66)for some d , as expressed in the basis ordered according to summands. The third summandin the decomposition is an irreducible eight-dimensional representation, which we express astwo basis entries. It follows thatΓ − V A Γ V = a a a
00 0 0 a ⊗ id × =: (cid:98) A ⊗ id × , (67)for some a , a , and a . Since id V ⊗ h V is not an intertwiner, the matrix Γ V ( id V ⊗ h V )Γ − V is not required to be diagonal. By first evaluating a partial trace, we havetr (cid:0) ( id V ⊗ h V ) δ V A (cid:1) = tr (cid:0) Γ V ( id V ⊗ h V )Γ − V ( (cid:99) δ V (cid:98) A ⊗ id × ) (cid:1) (68)= tr (cid:0) (cid:0) Γ − V ( id V ⊗ h V )Γ V (cid:1)(cid:99) δ V (cid:98) A (cid:1) . (69)Another computation shows that for V = X ( t ),tr (Γ − V ( id V ⊗ h V )Γ V ) = ∗ ∗ ∗ ∗∗ ∗ (70)and similarly for V = Y ( t ). If V = W ( t ), thentr (Γ − V ( id V ⊗ h V )Γ V ) = ∗ ∗ h
00 0 0 − h , (71)with h ∈ Q ( t ). Thus, tr (cid:0) Γ − V ( id V ⊗ h V )Γ V (cid:1)(cid:99) δ V (cid:98) A is traceless for any choice of V . This provesthe claim. (cid:3) Properties of ∆ g In this section, we prove several properties of ∆ g . In Corollary 6.2 we show that certainautomorphisms of g determine symmetries of ∆ g . If g is type A, D, E, G, or B even , we provein Lemma 6.3 that ∆ g it is preserved under the map t (cid:55)→ − t − . We then discuss theskein relation for ∆ sl , which we obtain from the characteristic polynomial of the R -matrix.We also discuss a method to compute the invariant on torus knots and give the formulafor (2 n + 1 ,
2) torus knots explicitly. We end this section by proving Theorem 1.2, that∆ sl dominates the Alexander polynomial for knots. More precisely, evaluating ∆ sl ( K ) at t = ± t = ± it − yields the Alexander polynomial in the variable t for any knot K . Lemma 6.1.
Let τ be a graph automorphism of the Dynkin diagram of g . Then τ determinesan automorphism of U Hζ ( g ) as a ribbon Hopf algebra.Proof. Let U h ( g ) be the topological quantum group over C [[ h ]] as defined in [CP95], howeverwe will denote its root vectors by E i and F i , rather than X ± i . Define (cid:98) τ h to be an algebraautomorphism of U h ( g ) so that (cid:98) τ h ( X i ) = X τ ( i ) for X ∈ { E, F, K } . Each of its Hopf algebramaps is intertwined by (cid:98) τ h . Let R h = e h (cid:80) (( A − ) ij H i ⊗ H j ) (cid:89) β exp q β ((1 − q − β ) E β ⊗ F β ) , (72)with the above product ordered according to the T i . In particular, R h is an R -matrix for U h ( g ) and (cid:98) τ h ⊗ (cid:98) τ h ( R h ∆( x )) = (cid:98) τ h ⊗ (cid:98) τ h (∆ op ( x ) R h ) (73) (cid:98) τ h ⊗ (cid:98) τ h ( R h )∆( (cid:98) τ h ( x )) = ∆ op ( (cid:98) τ h ( x )) (cid:98) τ h ⊗ (cid:98) τ h ( R h ) (74) (cid:98) τ h ⊗ (cid:98) τ h ( R h )∆( x ) = ∆ op ( x ) (cid:98) τ h ⊗ (cid:98) τ h ( R h ) . (75)By Proposition 8.3.13 in [CP95], this equality implies (cid:98) τ h ⊗ (cid:98) τ h ( R h ) = λR h (76)for some λ ∈ C [[ h ]] . Applying (cid:15) ⊗ id , we have λ ( (cid:15) ⊗ id ) R h = ( (cid:15) ⊗ id )( (cid:98) τ h ⊗ (cid:98) τ h ( R h )) (77) λ = (cid:98) τ h ⊗ (cid:98) τ h (( (cid:15) ⊗ id ) R h ) (78) λ = 1 . (79)Upon specialization of q to ζ , we define a surjection U h ( g ) → U Hζ ( g ) which maps R h to R by introducing the relations E α = F α = 0 for α / ∈ Φ + and E α = F α = 0 otherwise. Theaction of (cid:98) τ h descends to an automorphism (cid:98) τ on U Hζ ( g ). Note that the Cartan factor in both R h and R are easily seen to be preserved under (cid:98) τ h and (cid:98) τ , respectively. Under the quotientmap defined above, the non-Cartan factor in R is also preserved under (cid:98) τ . Thus, (cid:98) τ ⊗ (cid:98) τ ( R ) = R. Moreover, the induced action of τ on Φ + preserves the sum of all positive roots, and so thepivotal/balancing element is preserved. Thus, (cid:98) τ is an automorphism of U Hζ ( g ) as a ribbonHopf algebra. (cid:3) GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 19
Corollary 6.2.
Let τ be a graph automorphism of the Dynkin diagram of g . For any link L , τ determines a symmetry of the polynomial invariant: ∆ g ( L )( t , . . . , t n ) = ∆ g ( L )( t τ (1) , . . . , t τ ( n ) ) . (80) Proof.
By the above, precomposing with (cid:98) τ yields an automorphism of U H -mod as a ribboncategory. Let τ t denote ( t τ (1) , . . . , t τ ( n ) ). Then, this automorphism determines a mapEnd( V ( t )) (cid:98) τ −→ End( V ( τ t )) (81) λ ( t , . . . , t n ) id V ( t ) (cid:55)→ λ ( t τ (1) , . . . , t τ ( n ) ) id V ( τ t ) (82)and preserves scalars obtained from maps generated by elementary tangles. This impliesthat the invariant obtained from any link L satisfies∆ g ( L )( t , . . . , t n ) = ∆ g ( L )( t τ (1) , . . . , t τ ( n ) ) . (83) (cid:3) Lemma 6.3.
Let L be a link and suppose g is a Lie algebra of type A, D, E, G, or B even .Then ∆ g ( L )( t , . . . , t n ) = ∆ g ( L )( − t − , . . . , − t − n ) . (84) Proof.
Let L be a closed braid presentation of an oriented link colored by the representation V ( t ). We may compute ∆ g ( L ) directly from this braid presentation. However, ∆ g ( L ) canalso be computed after the isotopy of rotating the presentation of L by 180 ◦ . We now consider L to be colored by V ( t ) ∗ . By Proposition 3.4, L is colored by V ( − t − ). Thus, ∆ g ( L ) ispreserved under t (cid:55)→ − t − . (cid:3) The following is a corollary of the above symmetries and Theorem 1.2. This corollary is notused in the proof of Theorem 1.2.
Corollary 6.4.
The sl invariant is a Laurent polynomial in t and t .Proof. By the properties given in Corollary 6.2 and Lemma 6.3, ∆ sl is a linear combinationof symmetric Laurent polynomials. If ∆ sl ( L )( t , t ) is a polynomial whose degree in t is n ,then for some integers c, a i , b i , and c kl we have:∆ sl ( t , t ) = c + n (cid:88) i =1 (cid:2) a i ( t i t i − t − i t − i ) + b i ( t i t − i − t − i t i ) (cid:3) (85)+ n (cid:88) k =1 | l | 1) and ( t, − c kl , we have n (cid:88) i =1 (cid:2) a i ( t i − t − i ) + b i ( t i − t − i ) (cid:3) + n (cid:88) k =1 | l | Proposition 1.1. For all links L , ∆ sl ( L )( t , t ) is a Laurent polynomial in t and t suchthat ∆ sl ( L )( t , t ) = ∆ sl ( L )( t , t ) = ∆ sl ( L )( t − , t − ) . (1)Applying these symmetries of the unrolled sl invariant, we find that the polynomial containsredundant information. In Appendix 7, we provide only the information necessary to recoverit. Proposition 6.5. There is a nine-term skein relation for ∆ sl . GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 21 Proof. Let r be the 8 × r determines a relationamong powers of itself. Therefore, the characteristic polynomial of R t is the characteristicpolynomial of r raised to the power dim V ( t ). Thus, R t is a solution to the equation givenby r . This relation takes the form( R t + id )( t id + R t )( t R t + id )( t id + R t )( t R t + id )( t t id − R t )( t t R t − id ) = 0 . (95)After expansion and normalization, this implies the palindromic relation c R t + (cid:88) i =1 c i (cid:0) R i t + R − i t (cid:1) = 0 , (96)for some c , . . . , c ∈ Z [ t ± , t ± ]. Replacing each factor of R t with a diagrammatic strandcrossing and R t by two vertical strands, we obtain the skein relation. (cid:3) Similar to how we used the characteristic polynomial of the R -matrix to determine the skeinrelation, other characteristic polynomials yield relations among families of torus knots. Let q be a prime number, and r any positive integer less than q . Then for each 0 < n < q , wehave that qn + r and q are coprime. Define β q = (cid:32) q − (cid:89) i =0 id ⊗ i ⊗ R t ⊗ id ⊗ q − i − (cid:33) , (97)which acts on V ( t ) ⊗ q . Then the characteristic polynomial of β qq is some equation of the form q (cid:88) i =0 a i β qiq = 0 . (98)Multiplying this equation by β rq implies that the invariants of the torus knots of types( r, q ) , ( q + r, q ) , . . . , ((8 q − q + r, q ) determine the invariant for the (8 q q + r, q ) torus knot. Withthis information and after multiplying equation (98) by β r +1 q , we can deduce the invariant forthe ((8 q + 1) q + r, q ) torus knot and so on. This implies a recursion relation for all torus knots T nq + r,q , which can then be converted to an explicit function of n . The resulting expressionfor the q = 2 , r = 1 case is stated as a theorem below. Theorem 1.4 (Two Strand Torus Knots) . The value of ∆ sl on a (2 n + 1 , torus knot isgiven by: ( t − t − )( t n +21 + t − (4 n +2)1 )( t + t − )( t + t − )( t t − t − t − ) + ( t − t − )( t n +22 + t − (4 n +2)2 )( t + t − )( t + t − )( t t − t − t − )+ ( t t + t − t − )( t n +21 t n +22 + t − (4 n +2)1 t − (4 n +2)2 )( t t + t − t − )( t + t − )( t + t − ) . Remark 6.6. Observe that the expression for these torus knots breaks into three terms.One pair of terms exchange the roles of t and t , while the other is symmetric in t and t .We now move to the proof of Theorem 1.2, which will first require the following lemma. Lemma 6.7. Let → V → V → V → be an exact sequence of representations over a field and suppose f ∈ End ( V ⊗ n ) . Let π i ...i n ∈ End ( V ⊗ n ) denote the factor-wise projection or restriction to V i ⊗ · · · ⊗ V i n for each i ∈ { , , } n . Then tr ( f ) = (cid:88) ≤ i ,...,i n ≤ tr ( π i ...i n f π i ...i n ) , (100) Proof. Let f i ...i n denote π i ...i n f π i ...i n . We give a proof by induction. The base case is for theexact sequence itself, and is straightforward. We suppose the result holds through k = n − → V ⊗ V ⊗ n − → V ⊗ n → V ⊗ V ⊗ n − → f ∈ End( V ⊗ n ) we havetr( f ) = tr( f ... ) + tr( f ... ) (102)and the claim follows by the inductive hypothesis. (cid:3) Theorem 1.2 (Reduction to the Alexander-Conway Polynomial) . Let K be a knot. Then ∆ sl ( K )( t, ± 1) = ∆ sl ( K )( ± , t ) = ∆ sl ( K )( t, ± it − ) = ∆( K )( t ) . (2) Moreover, these are the only substitutions which yield the Alexander polynomial on all knots.Proof. Recall, by Propositions 3.8 and 3.10, that if V ( t ) has a four-dimensional irreduciblesubrepresentation V , then it fits into an exact sequence0 → V → V ( t ) → V → V . Let b ∈ B n be a braid whose closure is equal to K .We let b act on V ( t ) ⊗ n via the representation ψ n . Since b is associated to a knot, it followsfrom Schur’s lemma and Lemma 6.7 that we only need to consider endomorphisms of the n -fold tensor product of either V or V . More explicitly,tr (cid:0) ( id V ( t ) ⊗ h ⊗ n − V ( t ) ) ψ n ( b ) (cid:1) = tr (cid:0) ( id V ⊗ h ⊗ n − V ) ψ n ( b ) | V ⊗ n (cid:1) + tr (cid:0) ( id V ⊗ h ⊗ n − V ) ψ n ( b ) | V ⊗ n (cid:1) . (104)Then Theorem 1.7 implies,1dim V ( t ) tr (cid:0) ( id V ( t ) ⊗ h ⊗ n − V ( t ) ) ψ n ( b ) (cid:1) = 1dim V ( t ) (cid:0) dim V ∆( K )(( ξt ) ) + dim V ∆( K )( t ) (cid:1) (105)= dim V + dim V dim V ( t ) ∆( K )( t ) (106)= ∆( K )( t ) . (107)In the equation above, ξ = 1 and ξt is a component of the highest weight of V .The second claim of the theorem follows from checking which evaluations of ∆ sl simultane-ously yield the Alexander polynomial for the knots and . (cid:3) Note that this theorem only applies to knots. If a link has more than one component, thesimplification in equation (104) is no longer true, since each link component can be coloreddifferently. In which case, we must consider endomorphisms of (cid:78) ni =1 V j ( i ) for some non-constant j : { , . . . , n } → { , } . The example of the torus link T , given in equation (4) GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM GROUPS 23 of the introduction illustrates the dissimilarity between evaluations of ∆ sl and ∆ for a linkwith two components. 7. Values of ∆ sl In this section, we give the value of the unrolled restricted quantum sl invariant for allprime knots with at most seven crossings, as well as some higher crossing knots. We havereferred to [KA] for their list of prime knots and braid presentations. Among these examplesare knots that compare ∆ sl to other well-known invariants. The HOMFLY polynomial doesnot distinguish the knot n34 from n42 nor does it distinguish and , but ∆ sl does.The Jones polynomial differentiates and , but ∆ sl does not. The Jones polynomial andthe sl invariant both distinguish from ; however, the Alexander polynomial does not.Let L be a link. By Proposition 1.1, it is enough to specify the coefficient of t a t b in ∆ sl ( L )for each ( a, b ) in the cone C = { ( a, b ) ∈ Z | a ≥ | b | ≤ a } . (108)The coefficients of various knots can be found in Figures 8 and 9 below. We have boxedthe leftmost value on each cone, it has coordinates (0 , 0) and is the constant term in thepolynomial invariant for the indicated knot. We do not label zeros outside of the convexhull of nonzero entries in the cone. From the values given, we reconstruct ∆ sl by applyingreflections over the lines deg t = ± deg t . For example, the data for the trefoil knot is givenin Figure 8a and the associated Laurent polynomial is∆ sl ( )( t , t ) =( t t + t − t − ) − ( t t + t t + t − t − + t − t − ) + ( t + t + t − + t − )+ 2( t t + t − t − ) − t + t + t − + t − ) + ( t t − + t − t ) + 1 . Remark 7.1. One can find several patterns among these coefficients such as their shapesand alternating sign patterns, with some exceptions. A pattern among the knot invariantshere is that if the leading coefficient is 1, then the rightmost column gives the coefficientsof the Alexander polynomial. The i -th column of each computed invariant restricted todeg t ≥ t = deg t − i fordeg t ≤ 0. There is also a horizontal reflection symmetry among nonzero entries in eachcolumn restricted to deg t ≥ Remark 7.2. Using Theorem 1.2 and the symmetries of the invariant, we can recover thecoefficients of Alexander polynomial from these diagrams.We describe the method which corresponds to plugging in ± 1. Start at the coefficient of t j in the rightmost column, we will say it has coordinates ( n, j ). Consider the path in C ofline segments from ( n, j ) to (2 j, j ) to (2 j, − j ) to ( n, − j ). The sum of all terms alongthis path is the coefficient of t j in the Alexander-Conway polynomial. This is equivalent toconsidering all coefficients of the invariant, not the symmetry reduced form as in the figure,and adding all entries over column 2 j . The sum over a path starting from ( n, j + 1) is zero.Observe that along any of these paths starting at ( n, j + 1), there are an even numberof nonzero terms and that the second half of the sequence is given by multiplying the first − − 25 3 11 − − 11 1 − 21 1 − − − − − 37 10 66 − − 85 14 66 − − − 18 1 − − − − − − − 24 1 − − − − − − − − − 21 1 − − − − − − − − − 211 00 000 64 13 − − − 32 6 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Figure 8. The value of ∆ sl for all prime knots with fewer than seven crossings.half by − t and t t are opposite.The method associated to plugging in ± i is similar. If n is even, consider the path ofline segments from ( n, j ) to (cid:16) n − j, − ( n − j ) (cid:17) to ( n, j ). We take the alternating sumof terms along the path. 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