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A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3

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 sl2 at a fourth root of unity. In this context, the highest weight t ∈ C× of the representation determines the polynomial variable. For any semisimple Lie algebra g of rank n, we extend their construction to a link invariant ∆g, which takes values in n-variable Laurent polynomials. We prove general properties of these invariants, but the focus of this paper is the −1 case g = sl3. For any knot K, evaluating ∆sl3 at t1 = 1, t2 = 1, or t2 = it1 recovers the Alexander polynomial of K. We emphasize that this is not obvious from an examination of the R-matrix and that our proof requires several representation-theoretic results. We

tabulate ∆sl3 for all knots up to seven crossings along with various other examples. In particular, it distinguishes the Kinoshita–Terasaka knot and Conway knot mutant pair and is nontrivial on the Whitehead double of the trefoil.

1. Introduction Since the introduction of the Jones polynomial, an outstanding problem in quantum topology has been to give interpretations of quantum invariants of knots and 3-manifolds in terms of invariants from classical topology. Our motivating example is the Alexander-Conway polynomial, realized as the quantum invariant from unrolled restricted quantum sl2 at a primitive fourth root of unity. In [Mur92, Mur93, Oht02], Murakami and Ohtuski construct the Alexander-Conway polynomial from Turaev-type [Tur88] R-matrix actions on a family of 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 Uq(sl2), the variable in the Alexander-Conway polynomial is the parameter t ∈ C× of the unrolled restricted quantum group representation V (t). Unlike the standard representation of Uq(sl2), V (t) has quantum dimension zero; therefore, the naive R-matrix invariant assigns the value of zero to any closed tangle. Instead, using a modiﬁed trace, given by computing the invari- arXiv:2008.06983v2 [math.QA] 18 May 2021 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 are not as well understood as quantum invariants in rank one. The HOMFLY, Kauffman, and Kuperberg polynomials [Kau90, FHL+85, Kup94] are higher rank versions of the Jones polynomial. The Links-Gould invariants, among others, generalize the Alexander polynomial as an R-matrix invariant from quantum supergroups [LG92, KS91, GP07]. However, a higher rank version of the Alexander polynomial from unrolled restricted quantum groups has not been studied and is the subject of this paper. Specifically, we generalize the ADO invariants to higher rank at q2 = −1 [ADO92].

1 2 MATTHEW HARPER

Let ∆+ denote the positive roots of a Lie algebra g of rank n. Each character t, which × n ∼ + × we identify with (t1, . . . , tn) ∈ (C ) = Map(∆ , C ), determines a Verma module V (t) over the restricted quantum group U ζ (g), which is studied further in [Har19b]. We assume q = ζ is a primitive fourth root of unity and extend V (t) to a representation of the unrolled restricted quantum group. The associated quantum invariant ∆g is an assignment of a Lau- ± ± rent polynomial in Z[t1 , . . . , tn ] to every link L, and it is computed from a modiﬁed trace by coloring each component of L by V (t).

This invariant is not to be confused with the multivariable Alexander polynomial. In particular, the number of variables in ∆g depends on the rank of g and not on the number

of components of L. In fact, we compare ∆sl3 with the Alexander polynomial and other invariants in the Statement of Results. However, if L has m components, one can consider a modiﬁed version of ∆g by coloring each component of L by a distinct representation V (t). We discuss this multi-colored invariant brieﬂy, but our focus here is on the singly-colored invariant.

1.1. Statement of Results. The value of ∆sl3 on all prime knots up to seven crossings is tabulated in Figure 10. We have also computed this invariant for some higher crossing knots, allowing us to compare it with the Alexander, Jones, and HOMFLY polynomials. These values are found in Figure 11.

Most notable in these examples is that the Conway knot 11n34 and the Kinoshita-Terasaka

knot 11n42 have diﬀerent ∆sl3 polynomials. Therefore, this invariant can detect mutation. Our observation is consistent with the following result of Morton and Cromwell [MC96]: Colorings by representations with a multiplicity-free tensor product cannot detect mutation. The polynomial invariants of 11n34 and 11n42 are determined from Figure 1 below, as explained in Section 7.

In addition to 11n34 and 11n42, untwisted Whitehead doubles of knots have Alexander polynomial equal to 1 [Rol03]. It follows that the Alexander module of each of these knots is 0 zero. We take the Whitehead double of the trefoil Wh (31) as an example. In contrast to the

Alexander invariant, ∆sl3 assigns a non-trivial polynomial to these knots, see Figures 1 and 2.

The Alexander-Conway polynomial ∆ is dominated by ∆sl3 on knots and it is in the following

sense that ∆sl3 can be interpreted as a two-variable generalization of the classical knot invariant. Theorem 6.6 (Reduction to the Alexander-Conway Polynomial). Let K be a knot. Then −1 4 ∆sl3 (K)(t, ±1) = ∆sl3 (K)(±1, t) = ∆sl3 (K)(t, ±it ) = ∆(K)(t ). (88) Moreover, these are the only substitutions that yield the Alexander polynomial on all knots. Unlike the rank one case, this result cannot be deduced from the characteristic polynomial of the R-matrix for the speciﬁed parameters. These parameter values coincide precisely with the values of t ∈ (C×)2 for which V (t) is reducible. Suppose V (t) ﬁts into a short exact

sequence with submodule V1 and quotient V2. Then by naturality, ∆sl3 of a knot is equal to the invariants obtained from coloring the knot by either V1 or V2. For a multi-component link, Theorem 6.6 is false as only one component “changes color.” To prove the invariants A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 3

−4 −20 4 2 −4 8 8 −4 12 164 −46 12 −8 2 148 −34 476 −248 46 8 −4 0 496 −228 34 649 −476 164 4 −20 4 0 721 −496 148 −12 248 −46 −8 4 0 228 −34 12 −8 2 4

11n34 11n42

Figure 1. The value of ∆sl3 on the mutant pair 11n34 and 11n42.

−12 -23 12 0 Wh (31)

Figure 2. The value of ∆sl3 on the untwisted Whitehead double of 31.

obtained from single colorings by V1 or V2 are both related to the Alexander polynomial requires a separate argument. That result is stated below in Theorem 5.18.

We give an example of how Theorem 6.6 does not apply to links. We begin by stating the non-trivial fact that the multi-colored invariant of links is well-deﬁned [GPT09], which follows from the ambidexterity of V (t), i.e. the right and left partial quantum traces of any intertwiner on V (t) ⊗ V (t) are equal. We show that V (t) is ambidextrous in Theorem 5.10, which is based on work of Geer and Patureau-Mirand [GP13]. Further discussion on ambidexterity is given in Subsection 5.2. An important factor in the well-deﬁnedness of multi-colored link invariants is the Hopf link normalization, given here by: 2 −1 −1 −1 −1 ∆sl3 (21) = (t1 − t1 )(t2 − t2 )(t1t2 + t1 t2 ). (1) This normalization is analogous to the factor of (t − t−1)−1 considered when computing the multi-variable Alexander polynomial (Conway Potential Function) as a quantum invariant 2 [GPT09, Har19a, Mur93, Oht02]. However, normalization by ∆sl3 (21) does not lead to a specialization of the sl3 invariant of L to its Alexander polynomial. For example, let L = T4,2, 4 4 −4 be the singly-colored (4, 2) torus link. We see that ∆(T4,2)(t ) = t + t is not obtained from a “simple” evaluation of

∆sl3 (T4,2) 4 4 4 4 −4 −4 −4 −4 2 = t1t2 + t1 + t2 + t1 + t2 + t1 t2 . (2) ∆sl3 (21)

Theorem 6.6 could also be stated in terms of ∆sl3 and ∆sl2 , as ∆sl2 on knots equals the Alexander polynomial evaluated at t2. This motivates the following conjecture. Since the relation between invariants obtained from irreducible representations of algebras with diﬀer- ent ranks is not currently understood, this conjecture is not obvious and would require an eﬀective way of describing the actions of the R-matrix and pivotal element.

Conjecture 1.1. Let k be the Lie algebra obtained by removing m simple roots αj of a simply- × n laced Lie algebra g of rank n. Suppose the corresponding entries tj of t ∈ (C ) are chosen so 4 MATTHEW HARPER

ˆ × n−m that the representation V (t) of U ζ (g) has m maximal subrepresentations. Let t ∈ (C ) be m+1 obtained by removing the entries tj from t. Then for any knot K, ∆g(K)(t) = ∆k(K)(tˆ ).

The sl3 invariant admits a nine-term skein relation via the characteristic polynomial of the R-matrix represented in V (t) ⊗ V (t), see Proposition 6.9. Using a recursion determined by the square of the R-matrix, we have computed an explicit formula for (2n+1, 2) torus knots.

Theorem 6.10 (Two Strand Torus Knots). The value of ∆sl3 on a (2n + 1, 2) torus knot is given by:

−1 4n+2 −(4n+2) −1 4n+2 −(4n+2) (t1 − t1 )(t1 + t1 ) (t2 − t2 )(t2 + t2 ) −1 2 −2 −1 −1 + −1 2 −2 −1 −1 (t2 + t2 )(t1 + t1 )(t1t2 − t1 t2 ) (t1 + t1 )(t2 + t2 )(t1t2 − t1 t2 ) −1 −1 4n+2 4n+2 −(4n+2) −(4n+2) (t1t2 + t1 t2 )(t1 t2 + t1 t2 ) + 2 2 −2 −2 −1 −1 . (t1t2 + t1 t2 )(t1 + t1 )(t2 + t2 ) ∼ × 2 The following representation-theoretic discussion assumes g = sl3. Let P = (C ) denote the characters on the Cartan subalgebra of U ζ (g). Note that P has a group structure under entrywise multiplication with identity 1 = (1, 1). Lemma 1.2 ([Har19b]). The representation V (t) is reducible if and only if t belongs to any of 2 2 Xi = {t ∈ P : ti = 1} or X12 = {t ∈ P :(t1t2) = −1}. (3) + S Note that the Xα are indexed by the set of positive roots Φ . Let R = α∈Φ+ Xα and S Rα = Xα \ β∈Φ+\{α} Xβ . Deﬁnition 1.3. Let X(t),Y (t), and W (t) denote the head of V (t) for t belonging to exactly one of X2, X1, or X12, respectively.

ψ ψ For each ψ ∈ Ψ, let σ ∈ P be the weight of F v0 in V (1, 1). If t ∈ R, then V (t) is reducible and it fits into at least one exact sequence below: 0 → X(σ(001)t) →V (t) → X(t) → 0 (4) 0 → Y (σ(100)t) →V (t) → Y (t) → 0 (5) 0 → W (σ(010)t) →V (t) → W (t) → 0. (6) 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 to two of the above sequences, then t belongs to some pairwise intersection of X1, X2 and X12. In this case, V (t) has four, rather than two, composition factors in its Jordan-Hölderseries. Theorem 5.18 (Constructions of the Alexander Polynomial). The invariant of a link whose components are colored by a representation X(t), Y (t), or W (t) is the Alexander-Conway polynomial evaluated at t4. Like Theorem 6.6, this theorem cannot be proven directly from the skein relation determined by the characteristic polynomial of the R-matrix. Rather, we verify the Conway relation against an arbitrary tangle under the modified trace. The following tensor product decompositions play a key role in the argument. A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 5

Theorem 3.11 (Tensor Square Decompositions). For any t, s ∈ P such that the four- dimensional representations which appear are well-deﬁned and all summands are irreducible, the following isomorphisms hold:

X(t) ⊗ X(s) ∼= X(ts) ⊕ X(σ(110)ts) ⊕ V (σ(100)ts) (38) Y (t) ⊗ Y (s) ∼= Y (ts) ⊕ Y (σ(011)ts) ⊕ V (σ(001)ts) (39) W (t) ⊗ W (s) ∼= W (σ(100)ts) ⊕ W (σ(001)ts) ⊕ V (ts). (40) 1.2. Relation to Other Invariants. There are recent discoveries, similar in ﬂavor to the current work, relating invariants from the quantum supergroups glm|n and the Alexander polynomial. The Links-Gould invariants LGm,n are conjectured to satisfy the relation

LGm,n(L)(t, eiπ/m) = (∆(L)(t2m))n (7) 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 1.1 above.

Our observation that ∆sl3 assigns non-trivial polynomials to knots with trivial Alexander modules implies it is a non-abelian invariant in the sense of Cochran [Coc04]. Following

[Pic20], since ∆sl3 distinguishes 11n34 and 11n42, the invariant may contain information on sliceness. Nevertheless, we suspect ∆sl3 is related to other geometrically constructed invariants that are sensitive to knots with trivial Alexander modules. Knot Floer homology, for example, is non-trivial on the Whitehead double of 41 [Hed07]. Another example is the set of twisted Alexander polynomials for a particular matrix group [Wad94]. The set of twisted invariants derived from all parabolic SL2(F7) representations, up to conjugacy, of the knot groups of 11n34 and 11n42 are enough to distinguish the pair of mutant knots from each other and the unknot.

Another approach to refining Alexander invariants by passing to higher-rank Lie types are the SU(n) Casson invariants, developed by Frohman [Fro93]. However, these invariants for fibered knots are completely determined by their Alexander polynomials [BN00]. Since 51 and 10132 are fibered, their SU(n) Casson invariants are identical. These knots are distin- guished by ∆sl3 , demonstrating it is a stronger invariant on fibered knots.

It is also shown in [BCGP16] that the Reidemeister torsion is recovered from TQFTs based on the sl2 representations V (t). We expect that applying their TQFT to higher rank quantum groups at a fourth root of unity generalizes Reidemeister torsion and implies a Turaev surgery formula [Tur02] in terms of ∆g.

In rank one, Ohtsuki exhibits an isomorphism between the braid group representation determined 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 basis vectors of this decomposition to compute partial traces of intertwiners. This identiﬁcation recovers the determinant formula for the Alexander polynomial. Further investigation of the braid representations from X(t), Y (t), and W (t) may uncover a higher rank geometric construction of the Burau representation. This geometric interpretation could then extend to ∆sl3 . 6 MATTHEW HARPER

1.3. Further Questions. Here we give additional conjectures regarding the properties of the invariants ∆g and the representations studied in this paper. We have in Conjecture 1.1 above that if k ⊆ g then ∆g dominates ∆k.

Following Theorem 5.18, a result about links colored by a single 4-dimensional representation, preliminary computations suggest an extension to the multi-colored setting. For each family of representations {X(t)}, {Y (t)}, and {W (t)}, we claim that the relations for the Conway Potential Function, given in [Jia16], are satisﬁed. Conjecture 5.19. The multi-variable invariants obtained from links with components colored by a single palette {X(t)}, {Y (t)}, or {W (t)} are the Conway Potential Function.

−1 −1 In all known examples, we have found that ∆g(L)(t1, . . . , tn) = ∆g(L)(−t1 ,..., −tn ). The identity holds for invertible links by Lemma 6.3, however it has also been veriﬁed for the non-invertible knots 817, 932, and 933. In light of this and the results of Section 6, it is enough 2a 2b to specify the coeﬃcient of t1 t2 in ∆sl3 (L) for each (a, b) in the cone 2 C = {(a, b) ∈ Z |a ≥ 0 and |b| ≤ a} (8) to recover the polynomial invariant.

It is known that other quantum invariants such as the HOMFLY polynomial, and therefore the Jones and Alexander polynomials, cannot detect knot inversion. Therefore, we ask the following question. Question 6.4. Does there exist a link L and Lie algebra g such that −1 −1 ∆g(L)(t1, . . . , tn) 6= ∆g(L)(−t1 ,..., −tn )? (86)

Given that all known ∆sl3 polynomial invariants can be described by specifying their coeﬃ- cients on the cone C, we observe the following symmetry properties of these coeﬃcients.

Conjecture 7.1. The sl3 invariant satisfies the following properties. For all a, b ≥ 0: • If the leading coefficient is 1, then the rightmost nonzero column gives the coefficients of the Alexander-Conway polynomial • The coefficients in positions (a, b) and (a − b, −b) are equal if b is even and opposite if b is odd • The coefficients in positions (a, b) and (a, a − b) are equal if a is even and opposite if a is odd.

1.4. Structure of Paper. In Section 2 we recall the restricted quantum group U ζ (g). We also deﬁne a bilinear pairing on the subalgebra generated by F1,...,Fn, then prove it is non-degenerate.

We recall the representations V (t) in Section 3. The pairing deﬁned in the previous section ∗ ∼ −1 allows us to prove the isomorphism V (t) = V (−t ) in all Lie types. Specializing to sl3 and t ∈ Rα, we prove various results on the irreducible representations which appear in the composition series of V (t). Their relation to the Alexander polynomial is proven in Section 5.

Section 4 introduces the unrolled structure of U ζ (g) and derives the formula for the R- matrix from that of the topological quantum group based on various works [CP95, Lus90a, A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 7

Lus90b, Tan92]. The pivotal structure and R-matrix renormalization are given, allowing us to compute invariants from the representations V (t), rather than the unrolled quantum group H representations V (λ) deﬁned in this section. For g = sl3, we give the explicit R-matrix action on V (t) ⊗ V (t) expressed in terms of the direct sum decomposition from [Har19b].

We give an overview on computing invariants and the modiﬁed trace in Section 5. We discuss ambidexterity of V (t) and well-deﬁnedness of the unframed link invariant. We then prove that the four-dimensional representations X(t), Y (t), and W (t) yield the Alexander polynomial in the variable t4 for any link L.

Section 6 is concerned with the properties of ∆g such as symmetry and the eﬀect of orientation reversal. We also prove Theorem 6.6, describe the ∆sl3 skein relation, and a method to compute ∆sl3 for families of torus knots.

The value of ∆sl3 on knots up to seven crossings and several other examples is given in Section 7. We also make several observations regarding these polynomials and their presentation. 1.5. Acknowledgments. I am very grateful to Thomas Kerler for insightful discussions. I also thank Sergei Chmutov, Sachin Gautam, Simon Lentner, Vladimir Turaev and an anony- mous referee for their helpful comments and suggestions. I thank the NSF for partial support through the grant NSF-RTG #DMS-1547357.

Some results in Sections 6 and 7 rely on computations done in Maple 2018.0. The code can be downloaded from [H].

2. Restricted Quantum Groups

In this section, we recall the restricted quantum group U ζ (g) at a primitive fourth root of unity via generators and relations, based on [Lus90a, Lus90b]. We deﬁne a bilinear pairing on negative root vectors and prove in Corollary 2.6 that this pairing is non-degenerate. We will make use of this pairing in the next section to state a duality property on induced representations. Throughout this section, g is any semisimple Lie algebra and ζ is a ﬁxed primitive fourth root of unity, unless noted otherwise.

Let A denote the Cartan matrix of g, which is symmetrized by d. As described in [CP95], + we ﬁx an ordering

Suppose that α belongs to the Weyl group orbit of the simple root αi and di = 2. Then the root α is said to be negligible. Let Φ0 denote the set of negligible root vectors and + + + Φ := Φ \ Φ0 the positive roots which are not negligible. Deﬁne ∆ to be the simple roots in Φ+. We use the notation x − x−1 ζ = ζdβ and bxc = , (9) β d ζd − ζ−d

omitting subscripts when d = 1. We recall restricted quantum sl3 in the deﬁnition below. The restricted quantum groups U ζ (g) are given in [Lus90a, Lus90b]. 8 MATTHEW HARPER

Deﬁnition 2.1. The restricted quantum group U ζ (sl3) is the Q(ζ)-algebra generated by Eα,Fα,Kj for α ∈ {α1, α12, α2}, and 1 ≤ i, j ≤ 2 with relations: −1 KiKi = 1,KiKj = KjKi, (10) K E = ζAij E K ,K F = ζ−Aij F K , (11) i αj αi αj i i αj αi αj i [E ,F ] = δ bK c ,E2 = F 2 = 0, (12) αi αj ij i di α α We order Φ+ according to

α1

E12 = −(E1E2 + ζE2E1) and F12 = −(F2F1 − ζF1F2), (14) as deﬁned by Lusztig’s automorphisms Ti. Theorem 2.2 ([Len16]). We have the following identiﬁcations for non-simply-laced quantum groups: ∼ ×n ∼ U ζ (bn) = U ζ (a1 ), U ζ (cn) = U ζ (dn), (15) ∼ + ∼ + U ζ (f4) = U ζ (d4), U ζ (g2) = U ζ (a3) . (16)

+ + The g2 case is exceptional as E112 is a primitive generator, even though Φ = Φ . Therefore, + the structure of U ζ (g2) , the positive part of the quantum group with respect to the Borel decomposition, is determined by A3 Cartan data. Under this identiﬁcation, the quantum parameter is changed to ζ = −ζ.

For brevity, we will denote U ζ (g) by U when no confusion arises. There is Hopf algebra structure on U, but only the antipode will be used explicitly in this work: −1 −1 S(Ei) = −EiKi S(Fi) = −KiFi S(Ki) = Ki . (17) Our other conventions can be found in [Har19b]. Let Ψ denote the space of maps {0, 1}Φ+ . Theorem 2.3 ([Lus90b]). The restricted quantum group U has a PBW basis ψ ψ0 k 0 n {E F K : ψ, ψ ∈ Ψ and k ∈ Z }, (18) 0 ψ Q ψα ψ Q ψα k Qn ki where E = α∈Φ+ Eα , F = α∈Φ+ Fα , K = i=1 Ki , and the products are ordered with respect to

1−ψ2 ψ1 Lemma 2.5 ([Har19b]). Let ψ1, ψ2 ∈ Ψ such that ψ1 < ψ2. Then χ(1...1)(F F ) = 0. − 0 − Observe that each χψ determines a bilinear pairing on U . For each F,F ∈ U , we deﬁne 0 0 (F,F )ψ = χψ(FF ). A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 9

Corollary 2.6. The bilinear pairing (·, ·)(1...1) is non-degenerate. P ψ − Proof. Suppose X := aψF ∈ U is degenerate with respect to (·, ·)(1...1), i.e. hF,Xi = 0 − for all F ∈ U . Let aψ∗ be the nonzero coeﬃcient of greatest index in X. By Lemma 2.4, 1−ψ∗ ψ∗ ∗ 1−ψ∗ ψ (F ,F )(1...1) 6= 0. Using Lemma 2.5, for each ψ < ψ ,(F ,F )(1...1) = 0. Thus, 1−ψ∗ 1−ψ∗ ψ∗ (F ,X)(1...1) = aψ∗ (F ,F )(1...1) = 0. (20)

This contradicts that aψ∗ is nonzero. Thus, X = 0, which proves non-degeneracy of (·, ·)(1...1).

3. Representations of U ζ (g) Here we recall the representation V (t) as a Verma module over U. In Proposition 3.3, we use the non-degenerate pairing deﬁned in the previous section to show that V (t)∗ is isomorphic to V (−t−1) in all Lie types. We included these results for the convenience of the reader, as they will be used in Section 6. We then specialize to U ζ (sl3) for the remainder of this section, characterizing the structure of V (t) when it has a four-dimensional irreducible subrepresentation. Exact sequences in the diﬀerent cases are given in Propositions 3.7 and 3.10. In Theorems 3.11 and 3.12, we state the tensor product decompositions for these representations.

± ± 3.1. Induced Representations. Let P denote the characters on hK1 ,...,Kn i. Note that P has a group structure under entrywise multiplication with identity 1 = (1,..., 1). ∼ × n Moreover, P = (C ) under the identiﬁcation of each t ∈ P with its values on Ki. Let ± + B = hEα,Ki : α ∈ ∆ , 1 ≤ i ≤ ni be the Borel subalgebra. Each character t ∈ P extends to a character γt : B → C by

γt(Ki) = ti and γt(Ei) = 0. (21)

Deﬁnition 3.1. Let γt : B → C be a character as in (21). Let Vt = hv0i be the one-dimensional left B-module determined by γt, i.e. for each b ∈ B, bv0 = γt(b)v0. We deﬁne the representation V (t) to be the induced module U V (t) = IndB(Vt) = U ⊗B Vt. (22) Lemma 3.2. For every Lie algebra g, the lowest weight of V (t) is −t. Proof. First, suppose that g is of ADE type. As F (1...1) is the product of all positive root (1...1) (1...1) (2Aρ)i (1...1) vectors, F v0 is the lowest weight vector of V (t) and KiF v0 = ζ tiF v0, with the vector ρ denoting half the sum of positive root vectors expressed in the basis of simple roots αj. We refer to [Bou02] for the Cartan and root data. It is a routine computation to verify that ζ(2Aρ)i = −1 for all i and g of ADE type.

By Theorem 2.2, the type B, C, and F restricted quantum groups are isomorphic to one of simply-laced type. Since this isomorphism preserves rank, the previous argument applies.

+ + In type G2, since Φ = Φ we can compute the lowest weight using the usual Cartan data 2 −1 10 (2Aρ)i −3 2 and sum of positive roots [ 6 ]. Again, ζ = −1 for all i. Proposition 3.3. For every Lie algebra g and t ∈ P, V (t)∗ ∼= V (−t−1). 10 MATTHEW HARPER

Proof. We consider all types simultaneously. For each ψ ∈ Ψ, let fψ be the dual vector which ψ evaluates to one on F v0 and is zero otherwise. Observe that f(1...1) is a highest weight vector for V (t)∗. Indeed, for any v ∈ V (t),

−1 Ei · f(1...1)(v) = f(1...1)(−EiKi v) = 0, (23)

−1 (1...1) since −EiKi v cannot equal the lowest weight vector F v0.

ψ ∗ We claim that {F · f(1...1) : ψ ∈ Ψ} is a basis for V (t) . Let h·, ·i be the bilinear pairing 0 0 0 − ψ deﬁned as hF,F i = F · f(1...1)(F v0) for every F,F ∈ U . It follows that F · f(1...1) = P ψ ψ0 ψ ψ0∈ΨhF ,F ifψ0 . Moreover, {F · f(1...1) : ψ ∈ Ψ} is a basis if and only if h·, ·i is non- 0 0 degenerate. Recall the pairing (F,F )(1...1) = χ(1...1)(FF ) deﬁned in Section 2. For each ψ, ψ0 ∈ Ψ,

−1 ψ −ψ ψ0 ψ ψ ψ0 ψ ψ ψ0 hS (F )K ,F i = f(1...1)(K F F v0) = (−t) (F ,F )(1...1). (24)

−ψ −1 ψ −ψ − In the above, K is the product of Kα for which ψ(α) = 1 so that S (F )K ∈ U , ψ and (−t) is the analogous product of weights. The pairing (·, ·)(1...1) is non-degenerate by ψ ∗ ∗ ∼ Corollary 2.6, and so the dual vectors F · f(1...1) form a basis of V (t) . Thus, V (t) = V (s) for some s ∈ P.

−1 (1...1) Since Ki acts by Ki on the dual representation, s is inverse to the weight of F v0. By −1 Lemma 3.2, s = −t .

3.2. Specialization to Uζ (sl3). For the remainder of this section, we suppose g = sl3. By Theorem 2.3, we have that V (t) ∼= U − as vector spaces and

B = {1,F1,F2,F1F2,F12,F1F12,F12F2,F1F12F2} (25) = {F (000),F (100),F (001),F (101),F (010),F (110),F (011),F (111)} (26)

is an ordered basis of U −. Moreover, B determines the standard basis of V (t) by tensoring − basis vectors of U with v0.

We give the actions of E1 and E2 on the standard basis in Table 1 below. We also provide a graphical description of the action of U on V (t) in terms of weight spaces labeled by the standard basis in Figure 3. Each solid vertex indicates a one-dimensional weight space of V (t), and the “dotted” vertex indicates the two-dimensional weight space spanned by (101) (010) F v0 and F v0. An upward pointing edge is drawn between vertices if the action of either E1 or E2 is nonzero between the associated weight spaces. Downward edges are used to indicate nonzero matrix elements of F1 and F2. The green colored edges indicate actions of E1 and F1, and blue edges are for E2 and F2. For non-generic choices of the parameter t, edges are deleted from the graph because matrix elements of E1 and E2 vanish.

We now state the genericity condition on V (t). Let

2 2 2 X1 = {t ∈ P : t1 = 1}, X2 = {t ∈ P : t2 = 1}X12 = {t ∈ P :(t1t2) = −1}, (27) A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 11

Table 1. Actions of E1 and E2 on V (t) expressed in the standard basis. The remaining actions are zero for all t ∈ P.

(100) (000) (001) (000) E1F v0 = bt1c F v0 E2F v0 = bt2c F v0 (101) (001) (101) (100) E1F v0 = bζt1c F v0 E2F v0 = bt2c F v0 (010) (001) (010) −1 (100) E1F v0 = ζt1F v0 E2F v0 = −t2 F v0 (110) (101) (010) (011) −1 (101) (010) E1F v0 = ζt1F v0 − bζt1c F v0 E2F v0 = t2 F v0 + bt2c F v0 (111) (011) (111) (110) E1F v0 = bt1c F v0 E2F v0 = bt2c F v0

F1vh F2vh

F12vh F1F2vh

F1F12vh F12F2vh

F1F12F2vh

Figure 3. The action of U on the weight spaces of V (t).

then set R to be the union of X1, X2, and X12. We partition R into disjoint subsets indexed by nonempty subsets I ( Φ+, with ! ! \ [ RI = Xα \ Xα (28) α∈I α/∈I

Proposition 3.4 ([Har19b]). The representation V (t) of U is irreducible if and only if t ∈/ R.

If t belongs to R1, R2, or R12 then the socle of V (t) is a maximal subrepresentation of dimension four and it is also irreducible. Moreover, the head is four-dimensional, irreducible, and has highest weight t. We use Bi to denote the subalgebra of U generated by B and Fi.

3.3. The Representations X(t) and Y (t). We begin with t ∈ X1 and t ∈ X2 to deﬁne X(t) and Y (t), respectively.

X Deﬁnition 3.5. Suppose t ∈ X2. Let γt be the extension of the character γt on B to B2 X X with γt (F2) = 0. Set Xt = hx0i to be the one-dimensional B2-module determined by γt and deﬁne U X(t) = IndB2 (Xt) = U ⊗B2 Xt. (29)

The representation Y (t) is deﬁned analogously by assuming t ∈ X1 and setting the action of F1 to be zero on the generating vector. 12 MATTHEW HARPER

Remark 3.6. The representation X(t) is deﬁned if and only if t ∈ X2. Observe that

0 = [E2,F2]x0 = bK2c x0 = bt2c x0 (30) 2 if and only if t2 = 1. ψ ψ For each ψ ∈ Ψ, let σ ∈ P be the weight of F v0 in V (1, 1).

Proposition 3.7. If t ∈ X2 or t ∈ X1, we have the respective exact sequence 0 → X(σ(001)t) →V (t) → X(t) → 0 (31) 0 → Y (σ(100)t) →V (t) → Y (t) → 0. (32) As a subrepresentation of V (t), X(σ(001)t) has a basis given by (001) (101) (011) (111) hF v0,F v0,F v0,F v0i (33) and is indicated by the red points in Figure 4. The quotient representation is colored gray and the action of F2 which vanishes under the identiﬁcation is indicated by a dotted arrow. (001) Moreover, assuming t ∈ R2 is equivalent to assuming both X(σ t) and its quotient in V (t) are irreducible. Analogous statements are true for Y (σ(100)t) by switching the indices 1 and 2.

t ∈ R2 t ∈ R1

Figure 4. Reducible V (t) with subrepresentations X(σ(001)t) and Y (σ(100)t).

3.4. The Representation W (t). Motivating the t ∈ X12 case, we consider a quotient of V (t) so that there is a linear dependence between F1F2v0 and F2F1v0, i.e. F1F2v0−αF2F1v0 = 0 for some α ∈ Q(ζ, t). Then

E1(F1F2v0 − αF2F1v0) = 0,E2(F1F2v0 − αF2F1v0) = 0, (34) bζt c bt c and together imply α = 1 = 2 . These equalities involving α are true if and only if bt1c bζt2c 2 2 2 t1t2 = −1 and t1 6= 1. Therefore, we set

B12 = hB,F1F2 bK1c − F2F1 bζK1ci (35) W and let γt be the character on B12 which is an extension of γt on B and is zero otherwise.

Deﬁnition 3.8. Let t ∈ X12 and let Wt = hw0i be the one-dimensional B12-module deter- W mined by γt . We deﬁne W (t) by induction U W (t) = IndB12 (Wt) = U ⊗B12 Wt. (36) A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 13

Remark 3.9. To deﬁne W (t), we require t ∈ X12 so that E2.(F1F2 bK1c − F2F1 bζK1c)w0 vanishes. The dependence between F1F2w0 and F2F1w0 imply that W (t) is four dimensional. (010) We deﬁne an inclusion of W (σ t) in V (t) by sending w0 to (bt1c F1F2 − bζt1c F2F1) v0 and extending equivariantly. Quotienting out W (σ(010)t) returns us to the situation described in the motivation of this case.

Proposition 3.10. If t ∈ X12, we have the following exact sequence: 0 → W (σ(010)t) → V (t) → W (t) → 0. (37)

(010) In Figure 5, we assume t ∈ R12 so that both W (t) and W (σ t) are irreducible. Again, the subrepresentation is colored red and the resulting quotient is gray. Unlike Figure 4, the trivialized actions of F1 and F2 are not indicated by dotted arrows because the lowest (010) weight of W (t) is the same as the highest weight of W (σ t), and both F1 and F2 act non-trivially on this weight space in the subrepresentation.

t ∈ R12

Figure 5. Reducible V (t) with subrepresentation W (σ(010)t).

3.5. Tensor Decompositions Involving Irreducible Subrepresentations. We state two theorems on tensor decompositions of V (t), X(t), Y (t), and W (t) for sufficiently generic parameters. Theorem 3.11 (Tensor Square Decompositions). For any t, s ∈ P such that the four- dimensional representations which appear are well-defined and all summands are irreducible, the following isomorphisms hold: X(t) ⊗ X(s) ∼= X(ts) ⊕ X(σ(110)ts) ⊕ V (σ(100)ts) (38) Y (t) ⊗ Y (s) ∼= Y (ts) ⊕ Y (σ(011)ts) ⊕ V (σ(001)ts) (39) W (t) ⊗ W (s) ∼= W (σ(100)ts) ⊕ W (σ(001)ts) ⊕ V (ts). (40) Proof. To establish the isomorphism (38), we consider the module homomorphism f : V (ts) ⊕ V (σ(110)ts) ⊕ V (σ(100)ts) → X(t) ⊗ X(s) completely determined by the image of the highest weight vectors of V (ts), V (σ(110)ts), and V (σ(100)ts). We define the respective highest weight vectors in the image of f to be:

x0 ⊗ x0, ∆(E1E2E1).(F1F2F1x0 ⊗ F1F2F1x0), and ∆(E1).(F1x0 ⊗ F1x0). (41) (110) (100) Note that these have the correct weights. By assumption σ ts, σ ts ∈/ R1 ∪ R12, thus these vectors are nonzero and have distinct weights. By Proposition 3.7 and the remark following it, V (ts) and V (σ(110)ts) are reducible, but their heads are irreducible and have 14 MATTHEW HARPER

highest weights ts and σ(110)ts respectively. We also have that V (σ(100)ts) is irreducible by assumption. Thus, the head of each of V (ts), V (σ(110)ts), and V (σ(100)ts) is mapped to distinct nonzero subspaces under f. The socles of V (ts) and V (σ(110)ts) are irreducible and have highest weights σ(001)ts and −ts. Therefore, they must belong to ker f. Quotienting out this kernel yields the desired isomorphism.

We have a similar construction for (39), with the highest weight vectors given by exchanging indices 1 and 2 from (41). In the W (t) ⊗ W (s) case, the respective generating vectors are

∆(E1).(F1w0 ⊗ F1w0), ∆(E2).(F2w0 ⊗ F2w0) and w0 ⊗ w0. Although we will not use them in this paper, we include the data of mixed tensor products for completeness. Theorem 3.12 (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 (σ(010)ts) (42) X(t) ⊗ W (s) ∼= V (ts) ⊕ V (σ(100)ts) (43) Y (t) ⊗ W (s) ∼= V (ts) ⊕ V (σ(001)ts) (44)

V (t) ⊗ X(s) ∼= V (ts) ⊕ V (σ(100)ts) ⊕ V (σ(010)ts) ⊕ V (σ(110)ts) (45) V (t) ⊗ Y (s) ∼= V (ts) ⊕ V (σ(001)ts) ⊕ V (σ(010)ts) ⊕ V (σ(011)ts) (46) V (t) ⊗ W (s) ∼= V (ts) ⊕ V (σ(100)ts) ⊕ V (σ(001)ts) ⊕ V (σ(010)ts). (47) Proof. Using the same argument as above, we only provide highest weight vectors which generate an irreducible representation under the action of F1 and F2. We then check the weights of these generating vectors, which indicate the isomorphism class of the resulting representation. We omit cases below involving Y (t) when the result can be determined from the expression for X(t) by switching the indices 1 and 2.

X(t) ⊗ Y (s): x0 ⊗ y0, ∆(E1E2E1E2).(F1F2F1x0 ⊗ F2F1F2y0) X(t) ⊗ W (s): x0 ⊗ w0, ∆(E1).(F1x0 ⊗ F1w0) V (t) ⊗ X(s): v0 ⊗ x0, ∆(E1)(F1v0 ⊗ F1x0), ∆(E1E2E1E2)(F1F2F1v0 ⊗ F2F1F2x0), ∆(E1E2E1E2)(F1F2F1v0 ⊗ F1F2F1F2x0) V (t) ⊗ W (s): v0 ⊗ w0, ∆(E1).(F1v0 ⊗ F1w0), ∆(E2).(F2v0 ⊗ F2w0), ∆(E1E2E1E2).(F1F2v0 ⊗ F1F2F1F2w0)

4. Unrolled Restricted Quantum Groups and Braiding We begin this section by recalling the unrolled restricted quantum group in Definition 4.1. In Proposition 4.3, we show that its category of weight representations admits a braiding c. This braiding is obtained following results of [CP95, Tan92] in the h-adic case together with results of [Lus90a, Lus90b]. An outline of the proof precedes the proposition. We introduce the pivotal structure in this category in Subsection 4.2 and also provide a renormalization of the braiding that removes the dependence on the Hi-weights λ up to exponentiation. We A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 15 end this section with the renormalized action for quantum sl3 on the tensor decomposition of V (t) ⊗ V (t) in Corollary 4.7. Definition 4.1. Let g be a semisimple Lie algebra of rank n. We define unrolled re- H H stricted quantum g at a fourth root of unity U ζ (g), also denoted U , to be the algebra U ζ (g)[H1,...,Hn] with relations: ± ± HiKj = Kj Hi, [Hi,Ej] = AijEj, [Hi,Fj] = −AijFj, (48) in addition to the relations of U ζ (g) itself. The representation V (t) extends from the restricted quantum group to the unrolled restricted quantum group as follows. n λ Definition 4.2. Fix a character t ∈ P. Choose λ = (λ1, . . . , λn) ∈ C such that ζ = t, by H λi πiλi/2 H which we mean ζ := e = ti for each i ∈ {1, . . . , n}. We define V (λ) to be the U representation which restricts to V (t) on U and Hiv0 = λiv0 for each i ∈ {1, . . . , n}. H Since ti 6= 0 there are infinitely many isomorphism classes of representations V (λ) that restrict to the same V (t). In Proposition 4.5, we prove that the R-matrix action on V H (λ) ⊗ V H (λ), when properly normalized, depends only on t = ζλ.

H We say that a U -module V is a weight module if V is a direct sum of weight spaces, diλi the actions of H1,...,Hn are diagonal, and that Hiv0 = λiv0 implies Kiv0 = ζ v0. Let H H U -wmod denote the category of U -weight modules. The category U -wmod of weight H modules of U is obtained by forgetting the action of H1,...,Hn on U -wmod. 4.1. The R-matrix. A formula for the R-matrices of unrolled quantum groups at odd roots of unity is given in [GP13]. The formula is extended to general roots of unity in [CR21, Rup20] and agrees with our expression in (52). Here, we give more details on constructing this R-matrix based on Lusztig’s divided powers method for quantum groups at a root of unity [Lus90a, Lus90b] using an adapted proof from [CP95].

P −1 di(A )ij Hi⊗Hj We deﬁne the formal expression Eρ,ρ0 = ζ ij , which acts on weight represen- H tations V ⊗ W for representations (V, ρ), (W, ρ0) ∈ U -wmod as follows. Let v ∈ V and w ∈ W be weight vectors such that Hiv = µiv and Hjw = νjw, then let P −1 di(A )ij µiνj Eρ,ρ0 (v ⊗ w) = ζ ij (v ⊗ w). (49) H H H Let Ψζ be the automorphism of U ⊗ U deﬁned so that for all x, y ∈ U of weights α and β, respectively: −hα,βi −1 −1 Ψζ (x ⊗ y) = ζ (xKβ ⊗ yKα ). (50)

This automorphism is a specialization of Ψq as deﬁned in [CP95, Tan92] to q = ζ which acts on the unrolled restricted quantum group, also see [GP13, Theorem 40]. Similarly, Eρ,ρ0 is deﬁned in [CP95], but here is evaluated at q = ζ and acts on tensor product representations in H U -wmod. By the same computations given in [CP95, Proposition 10.1.19], Eρ,ρ0 implements H Ψζ on tensor products of highest weight representations, in the sense that for all x, y ∈ U : 0 −1 0 (ρ ⊗ ρ )(Ψζ (x ⊗ y)) = Eρ,ρ0 (ρ ⊗ ρ )(x ⊗ y)Eρ,ρ0 . (51) 16 MATTHEW HARPER

Before stating Proposition 4.3 and its proof, we give an overview here. The proposition is an analog of [CP95, Corollary 10.1.20], with the key diﬀerence being that we begin with div −1 Uq (g) as a Q[q, q ]-algebra, rather than Uq(g) over Q(q). This allows us to specialize q to div ζ by tensoring Q[ζ], which deﬁnes a homomorphism from Uq (g) to U ζ (g). We further use div div the fact that relations in the tensor completion of Uq (g) ⊗ Uq (g) are adapted verbatim in U ζ (g)⊗U ζ (g). The rest of the proof follows from computations involving Ψq of [CP95, Tan92] specialized to Ψζ .

0 H Proposition 4.3. Suppose (V, ρ), (W, ρ ) ∈ U ζ (g) -wmod and deﬁne   Y −1 cV,W = P ◦ Eρ,ρ0 (1 ⊗ 1 + (ζβ − ζβ )Eβ ⊗ Fβ) ∈ Hom H (V ⊗ W, W ⊗ V ) (52) U ζ (g) β∈Φ+ where P the tensor swap, Eρ,ρ0 deﬁned in the above sense as a linear map on tensor product H representations, and Eβ ⊗ Fβ acts in the usual sense. Then U ζ (g) -wmod is a braided H monoidal category with braiding c = {cV,W : V,W ∈ U ζ (g) -wmod}.

Proof. Let Uh(g) be the topological quantum group over C[[h]] with R-matrix deﬁned in [CP95]. Their formula for the R-matrix factors as

Y P −1 R = exp ((1 − q−2)E ⊗ F ), E = eh di(A )ij Hi⊗Hj ,R = R R , (53) EF qβ β β β h HH EF β with the above product deﬁned in the standard tensor completion over C[[h]] and ordered + with respect to the ordering on Φ . The REF term is expressible using Lusztig’s divided (n) n (n) n powers elements Eβ = Eβ /[n]dβ ! and Fβ = Fβ /[n]dβ !, ∞ 1 Y X 2 n(n−1) −1 n (n) (n) REF = qβ (qβ − qβ ) [n]dβ !Eβ ⊗ Fβ . β n=0

div −1 By [Lus90b, Theorem 6.7], Uq (g) is a subalgebra of Uh(g) over Q[q, q ] with basis given (n) (n) div div by the collection of Kβ and divided powers elements Eβ and Fβ . Let Uq (g)⊗Uq (g) div div −1 be the tensor product completion of Uq (g) ⊗ Uq (g) over Q[q, q ], which follows from the ˆ ˆ construction of Uq⊗ˆ Uq found in [CP95, Section 10.1.D] and [Tan92] where the ground ring div div is assumed to be Q(q). Then REF is well-deﬁned in Uq (g)⊗Uq (g).

As in the Lusztig construction [Lus90b], to implement the specialization to a root of unity we div −1 −1 2 −1 2 tensor Uq (g) as a Q[q, q ]-algebra with Q[v, v ]/(v + 1) and identify Q[v, v ]/(v + 1) div with Q[ζ]. This deﬁnes a homomorphism φ : Uq (g) → U ζ (g) which sends q to ζ, and div div deﬁnes a homomorphism on each graded component of the completion Uq (g)⊗Uq (g). Since φ([n]!) = φ([n]3!) = 0 for n ≥ 2 and φ([n]2!) = 0 for n ≥ 1, we have

Y −1 REF,ζ := φ ⊗ φ(REF ) = (1 ⊗ 1 + (ζβ − ζβ )Eβ ⊗ Fβ) ∈ U ζ (g) ⊗ U ζ (g). β∈Φ+

Observe that the automorphism Ψq deﬁned on Uq(g) ⊗ Uq(g) as in [CP95, Tan92] restricts div div div div to Ψq on Uq (g) ⊗ Uq (g) and (φ ⊗ φ) ◦ Ψq = Ψζ |U⊗U . Therefore, by [CP95, Lemma A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 17

10.1.18] and [Tan92], we have for all x ∈ U: op −1 Ψζ (∆ (x)) = REF,ζ ∆(x)REF,ζ (54)

(Ψζ )23((REF,ζ )13(REF,ζ )23) = ∆ ⊗ id(REF,ζ ) (55)

(Ψζ )12((REF,ζ )13(REF,ζ )12) = id ⊗ ∆(REF,ζ ). (56) The computations of [CP95, Proposition 10.1.19] are unchanged with the exception of the q specialization. It is easily veriﬁed that (54) is also satisﬁed for x = H1,...,Hn. Thus, H adapting [CP95, Corollary 10.1.20] shows that c is a braiding on U -wmod. Observe that the inverse braiding is

reverse  −1 Y −1 −1 cV,W =  (1 ⊗ 1 − (ζβ − ζβ )Eβ ⊗ Fβ)Eρ,ρ0  ◦ P, (57) β∈Φ+ Qreverse + with α∈Φ+ indicating the product is ordered with respect to the opposite ordering on Φ . 4.2. Pivotal Structure and Renormalized R-matrix. According to our argument for H constructing the R-matrix, the pivotal structure for Uh(g) also descends to U ζ (g) on repre- 1−r −1 sentations. It is implemented by K2ρ = K2ρ , as in [GP13] for r = 2, with 2ρ the sum of ∗∗ positive roots. We take the natural isomorphism ϕV : V → V to be the pivotal structure on ∗∗ the category of weight representations, which canonically identiﬁes evalv ∈ V with v ∈ V −1 and multiplies by K2ρ .

P −1 H −1 iπ/2 jk di(A )jkλj λk Let θV be the scalar action t2ρ e on V ∈ U ζ (g) -wmod, where λi are the H highest H-weights of V . We also denote θV by θλ when V = V (λ). This expression is motivated by the usual action of the ribbon element on weight representations of Uh(g). In Lemma 5.2, we express θV in terms of a partial trace on the braiding as given in [GP18, H Subsection 4.4], which then yields a ribbon structure on weight representations of U ζ (g).

H For V,W ∈ U -wmod, define a natural transformation H −1 RV,W = (θW ⊗ 1)cV,W . (58) H It is readily verified that RV,W satisfies the Yang-Baxter equation since cV,W satisfies the −1 H H H equation and θW is a scalar. If V = W = V (λ), we denote RV,W by R(λ,λ). In Proposition H λ 4.5, we prove that R(λ,λ) depends only on t = ζ and not λ itself. Since V (t) ⊗ V (t) is H H H canonically identified with V (λ) ⊗ V (λ) as vector spaces, we can interpret R(λ,λ) as an H endomorphism of V (t) ⊗ V (t) denoted Rt. Although c is a formal braiding in U -wmod, H RV,W is not since one of the hexagon identities is not valid.

ψ ψ 2 Recall that σ is the weight of F v0 in V (t). A pair of characters (t, s) ∈ P is called non-degenerate if V (σψts) is irreducible for each ψ ∈ Ψ. Theorem 4.4 ([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 M V (t) ⊗ V (s) ∼= V (σψts). (59) ψ∈Ψ 18 MATTHEW HARPER

0 n λ λ0 H H Proposition 4.5. Suppose λ, λ ∈ C are such that ζ = ζ = t. Then R(λ,λ) and R(λ0,λ0) deﬁne the same operator Rt ∈ End(V (t) ⊗ V (t)).

H Proof. We compute the action of R(λ,λ) directly. We assume that λ is generic so that (t, t) is a non-degenerate pair, as assumed in Theorem 4.4. The above isomorphism extends H H to V (λ) ⊗ V (λ) since the action of each Hi is completely determined on a summand by its highest weight vector. Therefore, Rt acts by a constant on each multiplicity-one summand and as an ampliﬁed n × n matrix on the multiplicity-n summands. To compute H these values, we consider the action of R(λ,λ) on the highest weight vector of each summand, (1...1) (1...1) ψ ∆(E F ).(v0 ⊗ F v0) for each ψ ∈ Ψ. Note that the isomorphism is expressed in H terms of t and not λ itself. Since R(λ,λ) is an intertwiner, H (1...1) (1...1) ψ (1...1) (1...1) H ψ R(λ,λ)∆(E F )(v0 ⊗ F v0) = ∆(E F )R(λ,λ)(v0 ⊗ F v0) P −1 (1...1) (1...1) −1 ( ij (A )ij Hi⊗Hj ) ψ =∆(E F )(θλ ⊗ 1) ◦ P ◦ ζ (v0 ⊗ F v0) P −1 (1...1) (1...1) −1 ( ij (A )ij λi⊗Hj ) ψ =∆(E F ) ◦ P ◦ (1 ⊗ θλ )ζ (v0 ⊗ F v0).

P −1 −1 ( ij di(A )ij λiHj ) ψ For each ψ ∈ Ψ, we compute the action of θλ ζ on F v0:

P −1 P −1 −1 ( ij di(A )ij λiHj ) ψ ( ij di(A )ij λi(Hj −λj )) ψ θλ ζ F v0 = t2ρζ F v0. P −1 P −1 Observe that ij di(A )ijλi(Hj −λj)Fkv0 = − ij(A )ijAkiλiFkv0 = −λkFkv0. Therefore, ! P −1 P −1 ( ij di(A )ij λiHj ) ψ − α ψ(α)λα ψ Y 1−ψ(α) ψ θλ ζ F v0 = t2ρζ F v0 = tα F v0. α (1...1) (1...1) ψ (1...1) (1...1) ψ It remains to compute ∆(E F ) ◦ P ◦ (v0 ⊗ F v0) = ∆(E F )(F v0 ⊗ v0) in (1...1) (1...1) ψ terms of ∆(E F )(v0 ⊗ F v0). However, these expressions will be independent of λ H since they do not involve any Hi. A computation for the R(λ0,λ0) action is identical and also given entirely in terms of t. Thus, Rt is well-deﬁned End(V (t) ⊗ V (t)). H H H H Remark 4.6. A similar computation shows that R(µ,λ)R(λ,µ) ∈ End(V (λ) ⊗ V (µ)) can be expressed in terms of ζλ and ζµ. The above arguments produce a well-deﬁned operator in End(V (ζλ) ⊗ V (ζµ)).

2 Corollary 4.7. Suppose that g = sl3 and that (t, t) ∈ P is non-degenerate. Under the tensor product decomposition of V (t) ⊗ V (t), we have

V (t) ⊗ V (t) Rt V (t) ⊗ V (t)

=∼ =∼ , (60)

L ψ 2 r⊗id8×8 L ψ 2 ψ∈Ψ V (σ t ) ψ∈Ψ V (σ t ) with r given by 2 2 2 2 0 −ξ −2 −2 −2 −2 diag(t1t2, −t2, −t1) ⊕ −ξ 0 ⊕ diag(−t1 , −t2 , t1 t2 ) (61) (1...1) (1...1) in the basis determined by the highest weight vectors ∆(E F )(v0 ⊗ F v0) for

F ∈ {1,F1,F2,F1F2,F2F1,F1F2F1,F2F1F2,F1F2F1F2}. A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 19

Proof. Continuing oﬀ the proof of Proposition 4.5 in the sl3 case, the action of Rt on the direct sum decomposition is given by ! (111) (111) ψ Y 1−ψ(α) (111) (111) ψ Rt∆(E F )(v0 ⊗ F v0) = tα ∆(E F ) ◦ P ◦ (v0 ⊗ F v0). (62) α

(111) (111) ψ (111) (111) ψ We compute ∆(E F ) ◦ P ◦ (v0 ⊗ F v0) = ∆(E F )(F v0 ⊗ v0) in terms of (111) (111) ψ ∆(E F )(v0 ⊗ F v0). Notice that multiplying F1, F12, and F2 together in any order yields F1F12F2 up to a factor of ζ. For ψ∈ / {(101), (010)}, the equality

(111) (111) ψ Y h(α)ψ(α) −ψ(α) (111) (111) ψ ∆(E F )(F v0 ⊗ v0) = (−1) tα ∆(E F )(v0 ⊗ F v0) α

is straightforward to verify by reordering F1F12F2 before expanding the coproducts on each side and recalling h is the height function. For example,

(111) (111) −1 ∆(E F1F12F2)(F1F12v0 ⊗ v0) = ζ∆(E F2F12)(−ζt1 F1F12v0 ⊗ F1v0) −1 (111) −1 −1 = t1 ∆(E F2)(−ζt1 t2 F1F12v0 ⊗ F12F1v0) −2 −1 (111) = −ζt1 t2 ∆(E F2)(−ζF1F12v0 ⊗ F1F12v0) −2 −1 (111) = −t1 t2 ∆(E F1F12F2)(v0 ⊗ F1F12v0).

(111) (111) (111) (111) Lastly, we consider ∆(E F )(F1F2v0 ⊗ v0) and ∆(E F )(F2F1v0 ⊗ v0). Note F1F12F2 = −F2F1F2F1 = −F1F2F1F2. We compute, −1 ∆(F2F1F2F1)(F1F2v0 ⊗ v0) = ∆(F2F1F2)(ζt1 F1F2v0 ⊗ F1v0) −1 −1 −1 = ζt1 ∆(F2)(F1F2F1F2v0 ⊗ F1v0 − t1 t2 F1F2v0 ⊗ F1F2F1v0) −1 −1 −2 −1 = −ζt1 t2 F1F2F1F2v0 ⊗ F2F1v0 − ζt1 t2 F2F1F2v0 ⊗ F1F2F1v0 −2 −2 + t1 t2 F1F2v0 ⊗ F2F1F2F1v0

∆(F1F2F1F2)(v0 ⊗ F2F1v0) = ∆(F1F2F1)(F2v0 ⊗ F2F1v0) −1 −1 = ∆(F1)(F2F1F2v0 ⊗ F2F1v0 + ζt2 t1 F2v0 ⊗ F2F1F2F1v0) −1 = F1F2F1F2v0 ⊗ F2F1v0 + t1 F2F1F2v0 ⊗ F1F2F1v0 −1 −1 + ζt1 t2 F1F2v0 ⊗ F2F1F2F1v0. This shows that

(111) (111) −1 −1 (111) (111) ∆(E F )(F1F2v0 ⊗ v0) = −ζt1 t2 ∆(E F )(v0 ⊗ F1F2v0), and by an identical computation with swapped indices,

(111) (111) −1 −1 (111) (111) ∆(E F )(F2F1v0 ⊗ v0) = −ζt1 t2 ∆(E F )(v0 ⊗ F2F1v0).

Q h(α)ψ(α) 1−2ψ(α) This gives the action of Rt on the direct sum by the formula α(−1) tα for Q ψ(α) 1−2ψ(α) ψ∈ / {(101), (010)} and α(−ζ) tα otherwise. Remark 4.8. The proofs of Proposition 4.5 and Corollary 4.7 nearly compute the R-matrix action for general g. However, the permutation action requires further investigation on the higher multiplicity spaces. 20 MATTHEW HARPER

5. Invariants from Unrolled Restricted Quantum Groups The goal of this section is to prove Theorem 5.18. We begin with our conventions for the Reshetikhin-Turaev functor [RT91, Tur94], then show that we obtain an unframed invariant H of oriented 1-tangles (or long knots) from ambidextrous weight representations of U . Col- orings by V (t) ∈ U -wmod also yield well-deﬁned morphisms and we prove in Theorem 5.10 that every irreducible V (t) is an ambidextrous object. Thus, we may compute invariants of links colored by weight representations of U via a modiﬁed trace [GPT09], as explained in Subsection 5.2. In Subsection 5.3, we consider the sl3 case and we use Z to denote an irreducible representation belonging to the set {X(t, 1),Y (1, t),W (ζt, t−1)}. For such Z, the associated quantum invariants satisfy an Alexander-Conway relation and therefore are equivalent to the single-variable Alexander-Conway polynomial. The result is stated formally in Theorem 5.18.

H 5.1. The Reshetikhin-Turaev Functor. Let V and W be a representations in U -wmod over a ﬁeld K. The Reshetikhin-Turaev functor assigns linear maps to tangles, and we use the convention that an upward pointing strand is the identity on V , and a downward ∗ pointing strand is the identity on V . We assign the braiding cV,W to overcrossings and −1 cW,V to undercrossings. Recall the evaluation and coevaluation maps, 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,” and these maps satisfy the relations

(idV ⊗ evV )(coevV ⊗ idV ) = idV = (evgV ⊗ idV )(idV ⊗ coev^V ) (63) and

(evV ⊗ idV ∗ )(idV ∗ ⊗ coevV ) = idV ∗ = (idV ∗ ⊗ evgV )(coev^V ⊗ idV ∗ ). (64)

∗ ∗ H H ∼ evV ∈ HomU (V ⊗ V, K) ∼ coevV ∈ HomU (K,V ⊗ V )

∗ ∗ ∼ ev ∈ Hom H (V ⊗ V , ) ∼ coev ∈ Hom H ( ,V ⊗ V ) gV U K ^V U K

Figure 6. Conventions for cups and caps.

−1 Let hV denote the action of the pivotal element K2ρ introduced in the previous section. ∗ Given any basis (ei) of V and corresponding dual basis (ei ), the above maps are deﬁned as ∗ ∗ ∗ ∗ evV (ei ⊗ ej) = ei (ej), evgV (ei ⊗ ej ) = ej (hV ei), (65) X ∗ X ∗ −1 coevV (1) = ei ⊗ ei , coev^V (1) = ei ⊗ hV ei, (66) i i

−1 and do not depend on the choice of basis. Moreover, evgV can by deﬁned as evV ∗ ◦(φV ⊗idV ∗ ), and coev^V as (idV ∗ ⊗ φV )coevV ∗ using the pivotal structure from Subsection 4.2.

H Let tr : EndU (V ) → K denote the canonical trace. The notation tri indicates the partial trace over the i-th tensor factor of an endomorphism of V ⊗n. A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 21

⊗n H Deﬁnition 5.1. For an intertwiner A ∈ EndU (V ), the n-th partial quantum trace of A is the intertwiner on V ⊗n−1 given by

trn((idV ⊗n−1 ⊗ hV )A) = (idV ⊗n−1 ⊗ evgV ) ◦ (A ⊗ idV ∗) ◦ (idV ⊗n−1 ⊗ coevV ), (67) which we also denote by the right partial quantum trace trR(A). The ﬁrst, or left, partial quantum trace trL(A) is deﬁned similarly, as follows: −1 trL(A) = tr1((hV ⊗ idV ⊗n−1 )A) = (evV ⊗ idV ⊗n−1 ) ◦ (idV ∗ ⊗ A) ◦ (coev^V ⊗ idV ⊗n−1 ). (68) n−1 Taking successive right quantum partial traces (trR) (A), also denoted tr2...n (idV ⊗ ⊗n−1 hV )A , acts by a scalar a on V , assuming V is irreducible. Since tr(hV ) = 0, n ⊗n ⊗n trR(A) = tr(hV A) = tr tr2...n(hV A) = a tr(hV ) = 0 (69) and n−1 ⊗n−1 tr(trR (A)) = tr (idV ⊗ hV )A = a tr(idV ) = a dim(V ). (70) We refer to the component of a link diagram, under the Reshetikhin-Turaev functor, in the tensor position which is not multiplied by hV in (70) as the cut or through-strand. We will also use cl to indicate the full closure of a braid or tangle diagram, as a topological operation, which yields a link. H Lemma 5.2 ([GP18]). There is a ribbon structure on U ζ (g) -wmod given by trR(cV,V ), P −1 −1 ij di(A )ij λiλj H which acts by θλ = t2ρ ζ on V (λ). H Proof. We compute the action of trR(cV H (λ),V H (λ)) on the highest weight vector v0 ∈ V (λ). Using notation of Proposition 4.3, REF,ζ acts as the identity on the vector v0 ⊗ v for every H H H v ∈ V (λ). We also have P ◦ Eρ,ρ = Eρ,ρ ◦ P on weight representations V (λ) ⊗ V (λ). We assume that coev(1) is a sum over a basis of weight vectors (ei) and e1 = v0. Thus, X ∗ trR(cV H (λ),V H (λ))v0 = (id ⊗ ev^V H (λ))(P ◦ Eρ,ρREF,ζ ⊗ idV H (λ)∗ )(v0 ⊗ ei ⊗ ei ) i X ∗ = (id ⊗ ev^V H (λ))(P ◦ Eρ,ρ ⊗ idV H (λ)∗ )(v0 ⊗ ei ⊗ ei ) i X ∗ = (id ⊗ ev^V H (λ))(Eρ,ρ ⊗ idV H (λ)∗ )(ei ⊗ v0 ⊗ ei ). i

Since Eρ,ρ acts diagonally, X ∗ ∗ −1 (id ⊗ ev^V H (λ))(ei ⊗ v0 ⊗ ei ) = (id ⊗ ev^V H (λ))(v0 ⊗ v0 ⊗ v0) = t2ρ v0, i P −1 di(A )ij λiλj and Eρ,ρ(v0 ⊗ v0) = ζ ij (v0 ⊗ v0), we have P −1 ∗ −1 ij di(A )ij λiλj trR(cV H (λ),V H (λ))v0 = (ev^V H (λ))(Eρ,ρ ⊗ idV H (λ)∗ )(v0 ⊗ v0 ⊗ v0) = t2ρ ζ v0. Because the scalar action on the dual of a generic representation is given by evaluating this expression at −λ, we may apply the results of [GP18, Subsection 4.4] which state that trR(cV,V ) is a ribbon structure.

Remark 5.3. Since θλ = trR(cV H (λ),V H (λ)), H −1 −1 trR(R(λ,λ)) = trR((1 ⊗ θλ )cV H (λ),V H (λ)) = θλ trR(cV H (λ),V H (λ)) = idV H (λ). (71)

Therefore, trR(Rt) = idV (t). 22 MATTHEW HARPER

5.2. Link Invariant and Ambidextrous Representations. In this subsection, we recall the notion of an ambidextrous representation. As described in [GPT09], these representations can be used to deﬁne nontrivial quantum invariants of framed links when the usual Reshetikhin-Turaev construction would otherwise yield zero. Here we describe how this construction extends to unframed links.

⊗n H Let A ∈ EndU (V ) be the intertwiner assigned to an (n, n)-tangle by the Reshetikhin- n−1 Turaev functor. In order for the scalar (trR) (A) for to be meaningful in the context of quantum link invariants, it should be independent of whether we took a combination of left or right partial traces of A. We say that V is ambidextrous if and only if trL(A) = trR(A) for any ⊗2 H A ∈ EndU (V ). The reader is referred to [GPT09] for further discussion on ambidexterity. H Since U ζ (g) -wmod is a ribbon Ab-category in the sense of [GPT09], there is a well-defined invariant of closed ribbon graphs colored by at least one ambidextrous irreducible representation [GPT09, Theorem 3], which we now describe. We consider an oriented framed link L as a colored ribbon graph with at least one component colored by an ambidextrous irreducible representation V . Cutting a strand of L colored by V yields a (1,1)-ribbon tangle T identified with an endomorphism hT iV of V via the Reshetikhin-Turaev functor using the conventions given in Subsection 5.1. Since V is irreducible, hT iV is a scalar multiple of the identity, and ambidexterity implies hT iV is independent of the cut point. Thus, hT iV is an invariant of L as a framed link, which we denote by F 0(L). Definition 5.4. An oriented framed link has the zero-sum property if the sum of the entries in each column of its linking matrix is zero. By symmetry of the linking matrix, a link has the zero-sum property if the sum of the entries in each row of the linking matrix is zero. Lemma 5.5. For each oriented unframed link L, there is a unique zero-sum link z(L) such that the underlying link is L. Moreover, if L and L0 are smoothly isotopic, then so are z(L) and z(L0). Proof. The link z(L) is determined up to isotopy by assigning a framing to each component of L. Since the linking numbers between any two distinct components of L are an invariant, these framings are uniquely determined. If there is a smooth isotopy between L and L0, then there is an ambient isotopy between them which extends to their specified framings. We consider the following transformation z as a map between braid diagrams and framed braid diagrams, as indicated below. To each signed crossing of a link diagram, we apply a Reidemeister I move of the opposite sign to the over-strand. We have positioned the twists H so that they are compatible with our definition of RV,V , Rt, and their inverses. It is easy to show that z extends to a map between braids and framed braids; moreover, we will not distinguish this extension from z itself.

7→ 7→ i j i j i j i j

Figure 7. Local transformation z deﬁned on elementary braid diagrams. A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 23

Recall that the (i, j) entry of the linking matrix can be computed from a blackboard framed diagram by adding the signs of all crossings where strand i crosses under strand j, see [Rol03, Deﬁnition 5.D.3(3)]. Note that this also includes the case i = j.

The first transformed diagram in Figure 7 contributes a factor of +1 to the linking matrix in position (j, j) and −1 in position (i, j). In the latter diagram, the contributions are −1 to (i, i) and +1 to (j, i). If i = j, then in either case the contribution to the linking matrix is clearly zero. Lemma 5.6. Suppose b is a braid, then the closure of z(b) is a zero-sum link. Proof. As indicated in the discussion above, each modified crossing in cl(z(b)) contributes either Ejj − Eij or Eji − Eii to the linking matrix, depending on its original sign. Here Eij is the matrix with 1 in position (i, j) and is zero otherwise. Also note that if i = j, the contribution is zero. Both Ejj − Eij and Eji − Eii have column sums equal to zero for all i and j, and the linking matrix is given by a sum of such matrices. Therefore, the column sums of the linking matrix of cl(z(b)) are all zero. Corollary 5.7. Suppose b is a braid with closure L. Then the framed link given by cl(z(b)) is isotopic to z(L). Proof. Since the transformations in Figure 7 are Reidemeister I moves, the link underlying the closure of z(b) is isotopic to L. The corollary then follows from Lemma 5.5. We can extend the invariant of framed links F 0 defined in [GPT09] to an invariant of unframed links using the map z. H Definition 5.8. Let V be an ambidextrous and irreducible weight representation of U . 0 Suppose that L is an unframed link colored by V . We define ∆V (L) = F (z(L)). ⊗n We define ψn(b) to be the action of b on V , where each braid group generator σi acts by H ⊗n RV,V , as in (58), in positions i and i + 1 of V . A simple argument is given in Subsection 4.2 to show that the renormalized braiding satisfies the Yang-Baxter relation. Therefore, ψn is a braid group representation. H Proposition 5.9. Let V be an ambidextrous and irreducible weight representation of U . For each unframed link L with braid representative b ∈ Bn, 1 ∆ (L) = tr (id ⊗ h⊗n−1)ψ (b). (72) V dim V V V n Proof. Since the closure of b is a presentation of L, cl(z(b)) is a presentation of z(L) by Corollary 5.7. Under the standard Turaev formalism, each modified crossing as given in H −1 Figure 7 is exactly RV,V = (θV ⊗ idV )cV,V or its inverse. Thus, the action of the modified 1 ⊗n−1 braid is identified with ψn(b). The modified trace is given by dim V tr (idV ⊗ hV )ψn(b) 0 and computes the invariant F (z(L)). The following is a straightforward adaptation of [GP13, Section 5.7]. Theorem 5.10. If V (t) is irreducible, then it is ambidextrous. Proof. We refer to [GP13] throughout this proof. The first hypotheses of their Theorem 36 are verified following their proof of Theorem 38, using r = 2 for our fourth root of 24 MATTHEW HARPER

unity case. That is, V (t) is an irreducible representation and the vectors w0 = v0 ⊗ f(1...1) 0 (1...1) (1...1) ∗ (1...1) (1...1) and w0 = t2ρ(F ⊗ F )w0 in V (t) ⊗ V (t) , together with F ,E ∈ U satisfy 0 0 (1...1) the following properties: hw, w i := evV (t)⊗V (t)∗ (w ⊗ w ) = 1, and both ∆(F )w0 and (1...1) 0 ∆(E )w0 are nonzero U-invariant vectors.

(1...1) 0 (1...1) It remains to show that the conditions, ker ∆(F ) ⊂ kerh·, w0i and ker ∆(E ) ⊂ 0 kerhw0, ·i, of Theorem 36(b) are also satisﬁed. Any vector that pairs non-trivially with w0 = (1...1) (1...1) (1...1) t2ρ(F v0) ⊗ (F f(1...1)) must be a multiple of w0 = v0 ⊗ f(1...1). Since ∆(F )w0 6= 0, (1...1) 0 ker ∆(F ) is contained in kerh·, w0i. The other case is straightforward to verify. Thus, [GP13, Theorem 38] extends to the fourth root of unity case. Remark 5.11. In [GPT09], three suﬃcient criteria for ambidexterity of a module V are

H given. One criteria is that the braiding on V ⊗V is central in EndU (V ⊗V ). If the braiding were central, then ∆V could not detect mutation [MC96, Theorem 5]. H ⊗n H By Proposition 4.5, the braid representation ψn : Bn → EndU (V (λ) ) depends only λ on t = ζ and defines a representation ψn in EndU (V (t)) with the same matrix elements by assigning Rt and its inverse to modified crossings. We have the following corollary and definition. 0 Corollary 5.12. Suppose λ, λ0 ∈ Cn are such that ζλ = ζλ = t and V H (λ) is ambidextrous. ± ± Then for all links L, ∆V H (λ)(L) ∈ Z[t1 , . . . , tn ] and ∆V H (λ)(L) = ∆V H (λ0)(L). Definition 5.13. Suppose t ∈ (C×)n such that V (t) is ambidextrous and t = ζλ for some n λ ∈ C . We define the invariant ∆g of unframed links colored by V (t) to be the map L 7→ ∆V H (λ)(L).

In light of this deﬁnition, we extend our use the notation ∆V (L) to include the invariant of a link L colored by an ambidextrous representation V ∈ U -wmod when it is well-deﬁned. For example, ∆g = ∆V (t). Remark 5.14. Recall that F 0 is an invariant of multi-colored framed links. With the appropriate normalizations, ∆g extends to an invariant of multi-colored links, where each color is a representation belonging to the set {V (t): t ∈ P \ R}.

5.3. The Alexander-Conway Polynomial from Representations of U ζ (sl3). Through- out this subsection, we assume g = sl3. We consider the invariant of unframed links colored −1 by some irreducible U ζ (sl3) representation Z ∈ {X(t, ±1),Y (±1, t),W (ζt, ±t )} and show that it agrees with the Alexander-Conway polynomial in each case. We prove that the Alexander-Conway identity holds by ﬁrst showing in Lemma 5.15 that it vanishes on all but one of the direct summands of Z ⊗Z. In Lemma 5.17, we show that the remaining summand does not contribute to the modiﬁed trace of any Z ⊗ Z intertwiner. Putting these together, the main result is formally proven in Theorem 5.18.

Let RZ denote the action of Rt on Z ⊗ Z as a subrepresentation of V (t) ⊗ V (t). Note that the matrix elements of RZ are expressible in terms of t. Since Z ⊗ Z is multiplicity free, Z is an ambidextrous representation. Following the arguments of Subsection 5.2, there is a well-deﬁned invariant of links colored by Z which evaluates to 1 on the unknot, and we denote it by ∆Z . Let −1 2 −2 δZ = RZ − RZ − (t − t )idZ⊗Z , (73) A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 25 which we identify with the Alexander-Conway skein relation given in Figure 8.

− = (t2 − t−2)

1 4 Figure 8. Alexander-Conway skein relation in the variable (t 2 )

Lemma 5.15. The action of δZ is zero on each 4-dimensional direct summand of Z ⊗ Z.

Proof. We ﬁrst consider the Z = X(t) case for t ∈ R2. There is a surjection from V (t) ⊗ V (t) to Z ⊗ Z determined by the quotient V (t) Z in each tensor factor. Al- though V (t) ⊗ V (t) does not decompose as a sum of irreducibles, Corollary 4.7 can still be applied to compute Rt acting on speciﬁc vectors in V (t) ⊗ V (t) for generic t, which then descend to vectors in Z ⊗ Z after specializing parameters. That is, Rt acts on v0 ⊗ v0 and 2 2 −2 ∆(E1E2E1).(F1F2F1v0 ⊗ F1F2F1v0) by t1t2 and −t1 , respectively. Setting t = (t, ±1) and taking the above quotient V (t) ⊗ V (t) Z ⊗ Z, these vectors are mapped to the highest weight vectors of the 4-dimensional summands of Z ⊗ Z indicated in Theorem 3.11. Then −1 2 −2 RZ − RZ acts by t − t on both of x0 ⊗ x0 and ∆(E1E2E1).(F1F2F1x0 ⊗ F1F2F1x0). Thus, δZ is zero on the corresponding 4-dimensional summands.

The Z = Y (t) case for t ∈ R1 is identical, except the indices 1 and 2 are switched. For Z = −1 W (ζt, ±t ), we take the vectors ∆(E1)(F1v0 ⊗ F1v0) and ∆(E2)(F2v0 ⊗ F2v0). Generically, 2 2 −2 2 2 Rt acts by −t2 and −t1, respectively. Therefore, RZ acts by −t and −(ζt) = t on the corresponding summands of Z ⊗ Z whose highest weight vectors are ∆(E1)(F1w0 ⊗ F1w0) and ∆(E2)(F2w0 ⊗ F2w0). This shows that δZ acts as zero on these summands. Remark 5.16. We consider Z ∈ {X(t, ±1),Y (±1, t),W (ζt, ±t−1)} in our computations, 0 −1 2 −2 and exclude Z of the form W (t, ±ζt ) because δZ0 = −2(t − t ) on the 4-dimensional 0 0 −1 summands of Z ⊗ Z . Replacing R with R in δZ0 resolves this discrepancy. Since we recover the Alexander polynomial from these representations, which does not distinguish mirror images, using either convention is consistent with Theorem 5.18.

Lemma 5.17. Let iU and πU denote normalized inclusion and projection maps of the 8- dimensional summand U into and out of Z ⊗ Z. The trace of πU ◦ (id ⊗ hZ ) ◦ iU is zero.

Proof. Since id⊗hV is not an intertwiner on Z ⊗Z, we compute its action on the summands explicitly. We express the image of (id ⊗ hZ ) ◦ iU in terms of the direct sum basis from Theorem 3.11 in order to evaluate the projection. It is presented so that composing with πU restricts to the ﬁrst term. For Z = X(t, 1), (id ⊗ hV ) ◦ iZ is the linear map determined by −ζ btc btc ∆(E ).(F x ⊗ F x ) 7→ ∆(E ).(F x ⊗ F x ) + ∆(F ).(x ⊗ x ) 1 1 0 1 0 t2 bζtc 1 1 0 1 0 t bζtc 1 0 0 −1 ∆(F E ).(F x ⊗ F x ) 7→ ∆(F E ).(F x ⊗ F x ) 1 1 1 0 1 0 t2 1 1 1 0 1 0 1 ∆(F E ).(F x ⊗ F x ) 7→ ∆(F E ).(F x ⊗ F x ) 2 1 1 0 1 0 t2 2 1 1 0 1 0 ζ btc ∆(F F E ).(F x ⊗ F x ) 7→ ∆(F F E ).(F x ⊗ F x ) 1 2 1 1 0 1 0 t2 bζtc 1 2 1 1 0 1 0 26 MATTHEW HARPER btc + ∆(F F F ).(x ⊗ x ) t bζtc bζt2c 1 2 1 0 0 2ζ btc + ∆(E E E ).(F F F x ⊗ F F F x ) t3 bζtc bζt2c 1 2 1 1 2 1 0 1 2 1 0 ζ bζtc 2ζ btc ∆(F F E ).(F x ⊗ F x ) 7→ ∆(F F E ) − ∆(F F E ) .(F x ⊗ F x ) 2 1 1 1 0 1 0 t2 btc 2 1 1 t2 bζtc 1 2 1 1 0 1 0 2 btc2 − ∆(F F F ).(x ⊗ x ) t bζt2c 1 2 1 0 0 2ζ − ∆(E E E ).(F F F x ⊗ F F F x ) t2 1 2 1 1 2 1 0 1 2 1 0 −1 ∆(F F F E ).(F x ⊗ F x ) 7→ ∆(F F F E ).(F x ⊗ F x ) 1 2 1 1 1 0 1 0 t2 1 2 1 1 1 0 1 0 1 ∆(F F F E ).(F x ⊗ F x ) 7→ ∆(F F F E ).(F x ⊗ F x ) 2 1 2 1 1 0 1 0 t2 2 1 2 1 1 0 1 0 ζ bζtc ∆(F F F F E ).(F x ⊗ F x ) 7→ − ∆(F F F F E ).(F x ⊗ F x ) 1 2 1 2 1 1 0 1 0 t2 btc 1 2 1 2 1 1 0 1 0 2ζ + ∆(F F E E E )(F F F x ⊗ F F F x ). bζt2c t3 2 1 1 2 1 1 2 1 0 1 2 1 0

Composing with the projection to U, we see that πU ◦ (id ⊗ hZ ) ◦ iU is traceless.

The cases for Z ∈ {X(t, −1),Y (±1, t)} are similar. The corresponding endomorphism on U = V (−t2, t−2) in the Z = W (it, t−1) case is determined from the mapping below, and the claim follows.

w0 ⊗ w0 7→ − w0 ⊗ w0 ζ bζtc 2ζ ∆(F ).(w ⊗ w ) 7→ ∆(F ).(w ⊗ w ) + ∆(E ).(F w ⊗ F w ) 1 0 0 btc 1 0 0 bt2c 1 1 0 1 0 ζ btc 2t ∆(F ).(w ⊗ w ) 7→ ∆(F ).(w ⊗ w ) + ∆(E ).(F w ⊗ F w ) 2 0 0 bζtc 2 0 0 bt2c 2 2 0 2 0 2 btc ∆(F F ).(w ⊗ w ) 7→∆(F F ).(w ⊗ w ) − ∆(F E ).(F w ⊗ F w ) 1 2 0 0 1 2 0 0 t bζtc bζt2c 2 1 1 0 1 0 2ζt + ∆(F E ).(F w ⊗ F w ) bζt2c 2 2 2 0 2 0 2ζt bζtc ∆(F F ).(w ⊗ w ) 7→∆(F F ).(w ⊗ w ) − ∆(F E ).(F w ⊗ F w ) 2 1 0 0 2 1 0 0 btc bζt2c 2 2 2 0 2 0 2 + ∆(F E ).(F w ⊗ F w ) t bζt2c 2 1 1 0 1 0 ζ bζtc 4ζ ∆(F F F ).(w ⊗ w ) 7→ − ∆(F F F ).(w ⊗ w ) − ∆(F F E ).(F w ⊗ F w ) 1 2 1 0 0 btc 1 2 1 0 0 t bt4c 1 2 1 1 0 1 0 ζ btc t ∆(F2F1F2).(w0 ⊗ w0) 7→ − ∆(F2F1F2).(w0 ⊗ w0) − ∆(F1F2E2).(F2w0 ⊗ F2w0) bζtc 2 btc2 bζtc2 A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 27

∆(F1F2F1F2).(w0 ⊗ w0) 7→ − ∆(F1F2F1F2).(w0 ⊗ w0) Theorem 5.18 (Constructions of the Alexander Polynomial). The invariant of a link whose components are colored by a representation X(t), Y (t), or W (t) is the Alexander-Conway polynomial evaluated at t4. Proof. Fix a four-dimensional representation Z to be any of X(t), Y (t), or W (t), and assume t is chosen so that Z is well-deﬁned and irreducible. Moreover, we may assume a speciﬁc form Z ∈ {X(t, ±1),Y (±1, t),W (ζt, ±t−1)} for some generic t, as explained in Remark 5.16. The Alexander-Conway relation is encoded by δZ and is zero on both 4-dimensional summands of Z ⊗ Z by Lemma 5.15. We next compute the quantum invariant for the closure of

H any intertwiner A ∈ EndU (Z ⊗ Z) with the skein relation δZ applied to it. In particular, we show that the closure of any (2,2)-tangle is compatible with the skein relation.

By Theorem 3.11, Z⊗Z is semisimple and multiplicity free. Therefore, δZ A acts by scalars on each summand and Lemma 5.15 states that only the action on the 8-dimensional summand

U may be nonzero. We denote this scalar by δdZ AU . The trace of (idZ ⊗ hZ )δZ A is computed in Figure 9, the dot indicates the application of hZ . In the ﬁrst equality, we sum over all summands of Z ⊗ Z, with the forks indicating projection and inclusion. All terms in the

sum are zero except the one which factors through U, on which, δZ A acts by δdZ AU . The diagram that remains is equal to the trace of πU ◦ (id ⊗ hZ ) ◦ iU , and by Lemma 5.17 this is zero.

δZ δZ A

= = δdZ AU = δdZ AU =0 A Σ U

U Z ⊗ Z Z ⊗ Z Z ⊗ Z

Figure 9. Proof of Theorem 5.18

We expect that our result extends to the multi-colored case, where all components are colored by the same family of representations. Conjecture 5.19. The multi-variable invariants obtained from links with components colored by a single palette {X(t)}, {Y (t)}, or {W (t)} are the Conway Potential Function.

6. Properties of ∆g

In this section, we prove several properties of ∆g. In Corollary 6.2 we show that certain automorphisms of g determine symmetries of ∆g. We prove in Lemma 6.3 that ∆g it is −1 preserved under the map t 7→ −t . We then discuss the skein relation for ∆sl3 , which we obtain from the characteristic polynomial of the Rt. We also discuss a method to compute the invariant on torus knots and give the formula for (2n + 1, 2) torus knots explicitly. We 28 MATTHEW HARPER

end this section by proving Theorem 6.6, that ∆sl3 dominates the Alexander polynomial for −1 knots. More precisely, evaluating ∆sl3 (K) at t2 = ±1 or t2 = ±it1 yields the Alexander 4 polynomial in the variable t1 for any knot K. Lemma 6.1. Let τ an automorphism of the Dynkin diagram of g. Then τ determines an H automorphism of U ζ (g) -wmod as a ribbon category. Proof. Recall the construction of the R-matrix for the unrolled restricted quantum group from the topological quantum group Uh(g) given in Proposition 4.3. Deﬁne τbh to be an algebra automorphism of Uh(g) so that τbh(Xi) = Xτ(i) for X ∈ {E,F,K,H}. The automorphism H τb, deﬁned similarly, acts on U ζ (g). Each of its Hopf algebra maps is intertwined by τbh. Let Rh be the R-matrix for Uh(g) given in (53). For each x ∈ Uh(g), op τbh ⊗ τbh(Rh∆(x)) = τbh ⊗ τbh(∆ (x)Rh) (74) op τbh ⊗ τbh(Rh)∆(τbh(x)) = ∆ (τbh(x))τbh ⊗ τbh(Rh) (75) op τbh ⊗ τbh(Rh)∆(x) = ∆ (x)τbh ⊗ τbh(Rh). (76) By Proposition 8.3.13 in [CP95], this equality implies

τbh ⊗ τbh(Rh) = λRh (77) for some λ ∈ C[[h]]. Applying ⊗ id, we have λ( ⊗ id)Rh = ( ⊗ id)(τbh ⊗ τbh(Rh)) (78) λ = τbh ⊗ τbh(( ⊗ id)Rh) (79) λ = 1, (80) as applying to either leg of the R-matrix yields 1 [CP95]. Thus, Rh is unchanged under τh ⊗ τh. It is easy to see that E is preserved by τh ⊗ τh, because τ preserves the Cartan ma- b b b b div trix. Therefore, REF is invariant under τbh ⊗ τbh. The subalgebra Uq (g) ⊆ Uh(g) is preserved div by τbh, so it admits a well-defined restriction to Uq (g) denoted τcdiv. The homomorphism div φ : Uq (g) → U ζ (g) defined in Proposition 4.3, intertwines the actions of τcdiv and τb, and implies τb ⊗ τb(REF,ζ ) = REF,ζ . H H Let τe : U ζ (g)-wmod → U ζ (g) -wmod be the functor defined by τe ((V, ρ)) = (V, ρ◦τb) on rep- H resentations and is the identity on morphisms as linear maps. That is, if F : U ζ (g)-wmod → Vect is the forgetful functor, then F ◦ τ = F. Since τ is a Hopf algebra morphism, τ is ∗e ∗ b e canonically a strict ⊗-functor and τe(V ) = τe(V ) up to canonical isomorphism. Therefore, τ(ev ) = ev up to canonical isomorphism and similarly for the other duality maps in e V τe(V ) Figure 6.

We prove τ(c ) = c for any weight representations (V, ρ) and (W, ρ0), noting that e V,W τe(V ),τe(W ) ρ and ρ0 are suppressed in our notation for the braiding. Since F is injective on morphisms, it is enough to show that F(τ(c )) = F(c ), which is the same as showing F(c ) = e V,W τe(V ),τe(W ) V,W F(c ). For this proof and its corollary, we distinguish the braiding c as an abstract τe(V ),τe(W ) V,W H morphism in U -wmod from the linear map realizing it. To be more precise, the realization given in (52) is, in fact, F(cV,W ). Since τb ⊗ τb(REF,ζ ) = REF,ζ , we have: 0 0 F(c ) = P ◦ E 0 ◦ (ρ ◦ τ ⊗ ρ ◦ τ)(R ) = P ◦ E 0 ◦ (ρ ⊗ ρ )(R ). (81) τe(V ),τe(W ) ρ◦τ,ρb ◦τb b b EF,ζ ρ◦τ,ρb ◦τb EF,ζ A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 29

0 Suppose that ρ(Hi)v = µiv and ρ (Hi)w = νiw, then ρ ◦ τ(Hi)v = ρ(Hτ(i))v = µτ(i)v and 0 b similarly ρ ◦ τe(Hi)w = ντ(i)w. Therefore, P −1 di(A )ij µ ν E 0 (v ⊗ w) = ζ ij τ(i) τ(j) (v ⊗ w) (82) ρ◦τ,ρb ◦τb P −1 ij di(A ) −1 −1 µiνj = ζ τ (i)τ (j) (v ⊗ w) = Eρ,ρ0 (v ⊗ w) (83) by invariance of the Cartan matrix under τ. Continuing from (81), 0 F(c ) = P ◦ E 0 ◦ (ρ ⊗ ρ )(R ) = F(c ). τe(V ),τe(W ) ρ◦τ,ρb ◦τb EF,ζ V,W Thus, τ(c ) = c . e V,W τe(V ),τe(W ) H In Proposition 5.2, we expressed the ribbon structure of U ζ (g) -wmod in terms of the braiding and pivotal action by tr (c ) = θ . Therefore, τ(θ ) = θ and τ is an automorphism R V,V V e V τe(V ) e H of U ζ (g) -wmod as a ribbon category. Corollary 6.2. Let τ be an automorphism of the Dynkin diagram of g. For any link L, τ determines a symmetry of the polynomial invariant:

∆g(L)(t1, . . . , tn) = ∆g(L)(tτ(1), . . . , tτ(n)). (84)

H Proof. As above, τ is the automorphism of U -wmod as a ribbon category given by pre- e 0 composing with τb. Moreover, τ induces a functor τe on U -wmod which is intertwined with τ by the functor that forgets the actions of H1,...,Hn. Let τ t denote (tτ(1), . . . , tτ(n)). If e 0 v0 is the highest weight vector in V (t), then ρ ◦ τ (Ki)v0 = ρ(Kτ(i))v0 = tτ(i)v0. Thus, 0 b τe (V (t)) = V (τ t).

Let hT iV (t) denote the action of a (1,1)-tangle representative of a zero-sum framed link L as an endomorphism of an irreducible representation (V (t), ρ). Since hT iV (t) is given by a composition of normalized braidings, evaluations, and co-evaluations, Lemma 6.1 implies 0 τ (hT i ) = hT i 0 = hT i . Applying the forgetful functor F, we have the equal- V (t) τe (V (t)) V (τ t) e 0 ity of linear maps F(hT iV (t)) = F(τe (hT iV (t))) = F(hT iV (τ t)). Since F(hT iV (t)) is equal to ∆g(L)(t1, . . . , tn) times the identity, the equality in (84) holds. Lemma 6.3. Let L be an oriented link and −L the same link with all orientations reversed. Then −1 −1 ∆g(−L)(t1, . . . , tn) = ∆g(L)(−t1 ,..., −tn ). (85) Proof. Recall from Proposition 3.3 that V (t)∗ ∼= V (−t−1), and by Theorem 5.10 both V (t) and its dual are ambidextrous for generic t. Since the morphism assigned to the open Hopf link colored by V (t) and V (t)∗ is nonzero, we may apply [GPT09, Proposition 19]. Thus, reversing the orientation of a component of L is equivalent to coloring it by V (t)∗. Therefore, −1 ∆g(−L) is computed from coloring all components of L by V (−t ). −1 −1 In all known examples, we have found that ∆g(L)(t1, . . . , tn) = ∆g(L)(−t1 ,..., −tn ). In particular, this equality holds for the non-invertible knots 817, 932, and 933. Question 6.4. Does there exist a link L and Lie algebra g such that −1 −1 ∆g(L)(t1, . . . , tn) 6= ∆g(L)(−t1 ,..., −tn )? (86) 30 MATTHEW HARPER

Applying these symmetries of the unrolled sl3 invariant, we ﬁnd that the polynomial contains redundant information. In Section 7, our presentation of the knot invariants accounts for these symmetries.

Lemma 6.5. Suppose that t ∈ R so that V (t) belongs to the exact sequence

0 → V1 → V (t) → V2 → 0, (87)

where V1 and V2 are ambidextrous. Then for any knot K, ∆V (t)(K) = ∆V1 (K) = ∆V2 (K).

Proof. Given that V1 and V2 are ambidextrous, ∆V1 and ∆V2 are well-defined. Let b ∈ Bn be a braid representative for a knot K, and T the (1,1)-tangle obtained from closing the n − 1 right strands of z(b), with z as in Figure 7. Recall that b acts on V (t)⊗n via the n−1 representation ψn given before Definition 5.13. For generic t, hT iV (t) = trR (ψn(b)) acts on V (t) by the scalar ∆V (t)(K). We specialize this generic t so that it belongs to R and V (t) is reducible. Then hT iV (t) still acts by a multiple of the identity since specifying values of t is done by direct substitution. By naturality of the braiding and pivotal structure discussed in Section 4, the inclusion i : V1 ,→ V (t) satisfies the intertwiner relation

∆V (t)(K) · i = hT iV (t) ◦ i = i ◦ hT iV1 = ∆V1 (K) · i.

Therefore, ∆V1 (K) = ∆V (t)(K).

Similarly, the surjection V (t) V2 intertwines the scalar action. This implies that hT iV2 acts on V2 by ∆V2 (K) = ∆V (t)(K).

Theorem 6.6 (Reduction to the Alexander-Conway Polynomial). Let K be a knot. Then

−1 4 ∆sl3 (K)(t, ±1) = ∆sl3 (K)(±1, t) = ∆sl3 (K)(t, ±it ) = ∆(K)(t ). (88)

Moreover, these are the only substitutions that yield the Alexander polynomial on all knots.

Proof. The ﬁrst claim is an immediate consequence of Theorem 5.18 and Lemma 6.5. The second claim follows from checking which evaluations of ∆sl3 simultaneously yield the Alexan- der polynomial for the knots 31 and 41. Remark 6.7. Lemma 6.5 only applies to knots. If a link were colored by reducible representations V (t), only the color of the open strand could be replaced by V1 or V2. All other strands in the diagram remain colored by V (t).

The following is a corollary of the above symmetries and Theorem 6.6.

Corollary 6.8. Suppose that for some link L, ∆sl3 (L) = ∆sl3 (−L). Then ∆sl3 (L)(t1, t2) is 2 2 a Laurent polynomial in t1 and t2.

Proof. By the properties given in Corollary 6.2 and our assumption, ∆sl3 (L)(t1, t2) is a linear i j i+j −i −j combination Laurent polynomials which are symmetric under t1t2 7→ (−1) t1 t2 and i j j i t1t2 7→ t1t2. If ∆sl3 (L)(t1, t2) is a polynomial whose degree in t1 is n, then for some integers A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 31 c, ai, bi, and ckl we have: n X i i −i −i i −i −i i ∆sl3 (t1, t2) = c + ai(t1t2 + t1 t2 ) + bi(t1t2 + t1 t2) i=1 n X k l k+l −k −l l k k+l −l −k + ckl t1t2 + (−1) t1 t2 + t1t2 + (−1) t1 t2 . k=1 |l|

This implies that ckl = 0 whenever k and l have diﬀerent parity.

We now consider the evaluations of ∆sl3 at (t, 1) and (t, −1). Only accounting for terms where k and l have the same parity, we have n n X i −i i −i X k −k l −l ai(t + t ) + bi(t + t ) + ckl t + t + t + t (91) i=1 k=1 |l|

6.1. Properties Derived from Powers of Rt. The skein relation and value of ∆sl3 on two strand torus knots are both derived from the characteristic (minimal) polynomial of Rt. The former is obtained from (93), and the latter is stated in Theorem 6.10.

Proposition 6.9. There is a nine-term skein relation for ∆sl3 . Proof. Let r be the 8 × 8 matrix which appears in Corollary 4.7. By the Cayley-Hamilton Theorem, the characteristic polynomial of r determines a relation among powers of itself. Therefore, the characteristic polynomial of Rt is the characteristic polynomial of r raised to the power dim V (t). Thus, Rt is a solution to the equation given by r. This relation takes the form 2 2 2 2 2 2 2 2 2 (Rt + id)(t1id + Rt)(t1Rt + id)(t2id + Rt)(t2Rt + id)(t1t2id − Rt)(t1t2Rt − id) = 0. (93) 32 MATTHEW HARPER

After expansion and normalization, this implies the palindromic relation 4 0 X i −i c0Rt + ci Rt + Rt = 0, i=1 ±2 ±2 for some c0, . . . , c4 ∈ Z[t1 , t2 ] determined by (93). Replacing each factor of Rt with a 0 diagrammatic strand crossing and Rt by two vertical strands, we obtain the skein relation. Similar to how we used the characteristic polynomial of the R-matrix to determine the skein relation, 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, we have that qn + r and q are coprime. Deﬁne q−2 ! Y ⊗i ⊗q−i−2 βq = id ⊗ Rt ⊗ id , (94) i=0 ⊗q q which acts on V (t) . Then the characteristic polynomial of βq is some equation of the form 8q X qi aiβq = 0. (95) i=0 r Multiplying this equation by βq implies that the invariants of the torus knots of types (r, q), (q+r, q),..., ((8q−1)q+r, q) determine the invariant for the (8qq+r, q) torus knot. With r+1 this information and after multiplying equation (95) by βq , we can deduce the invariant for the ((8q +1)q +r, q) torus knot and so on. This implies a recursion relation for all torus knots Tnq+r,q, which can then be converted to an explicit function of n. The resulting expression for the q = 2, r = 1 case is stated as a theorem below.

Theorem 6.10 (Two Strand Torus Knots). The value of ∆sl3 on a (2n + 1, 2) torus knot is given by: −1 4n+2 −(4n+2) −1 4n+2 −(4n+2) (t1 − t1 )(t1 + t1 ) (t2 − t2 )(t2 + t2 ) −1 2 −2 −1 −1 + −1 2 −2 −1 −1 (t2 + t2 )(t1 + t1 )(t1t2 − t1 t2 ) (t1 + t1 )(t2 + t2 )(t1t2 − t1 t2 ) −1 −1 4n+2 4n+2 −(4n+2) −(4n+2) (t1t2 + t1 t2 )(t1 t2 + t1 t2 ) + 2 2 −2 −2 −1 −1 . (t1t2 + t1 t2 )(t1 + t1 )(t2 + t2 ) Remark 6.11. Observe that the expression for these torus knots breaks into three terms. One pair of terms exchange the roles of t1 and t2, while the other is symmetric in t1 and t2.

7. Values of ∆sl3

In this section, we give the value of the unrolled restricted quantum sl3 invariant for all prime knots with at most seven crossings, as well as some higher crossing knots. We have referred to [KA] for their list of prime knots and braid presentations. Among these examples are knots that compare ∆sl3 to other well-known invariants. The HOMFLY polynomial does not distinguish the knot 11n34 from 11n42 nor does it distinguish 51 and 10132, but ∆sl3 does.

The Jones polynomial diﬀerentiates 61 and 946, but ∆sl3 does not. The Jones polynomial and the sl3 invariant both distinguish 89 from 10155; however, the Alexander polynomial does not. A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 33

As indicated by Question 6.4, ∆sl3 (L) = ∆sl3 (−L) in all known examples. By the results of 2a 2b Section 6, it is enough to specify the coeﬃcient of t1 t2 in ∆sl3 (L) for each (a, b) in the cone 2 C = {(a, b) ∈ Z |a ≥ 0 and |b| ≤ a}. (96) The coeﬃcients of various knots can be found in Figures 10 and 11 below. We have boxed the leftmost value on each cone, it has coordinates (0, 0) and is the constant term in the polynomial invariant for the indicated knot. We do not label zeros outside of the convex

hull of nonzero entries in the cone. From the values given, we can reconstruct ∆sl3 since the coeﬃcient in position (a, b) is equal to those in positions (b, a), (−a, −b), and (−b, −a). For example, the data for the trefoil knot 31 is given in Figure 10 and the associated Laurent polynomial is 4 4 −4 −4 4 2 2 4 −4 −2 −2 −4 4 4 −4 −4 ∆sl3 (31)(t1, t2) =(t1t2 + t1 t2 ) − (t1t2 + t1t2 + t1 t2 + t1 t2 ) + (t1 + t2 + t1 + t2 ) 2 2 −2 −2 2 2 −2 −2 2 −2 −2 2 + 2(t1t2 + t1 t2 ) − 2(t1 + t2 + t1 + t2 ) + (t1t2 + t1 t2) + 1.

The following properties are true in all computed examples, but they have not been proven.

4j We describe the method which corresponds to plugging in ±1. Start at the coefficient of t2 in the rightmost column, we will say it has coordinates (n, 2j). Consider the path in C of line segments from (n, 2j) to (2j, 2j) to (2j, −2j) to (n, −2j). The sum of all terms along this path is the coefficient of tj in the Alexander-Conway polynomial. This is equivalent to considering all coefficients of the invariant, not the symmetry reduced form as in the figures, and adding all entries over column 2j. The sum over a path starting from (n, 2j + 1) is zero. More specifically, the cancellations occur between pairs of terms according to Conjecture 7.1.

−1 The method of gathering terms after evaluating t2 at ±ζt is similar. If n is even, con- n n1 sider the path of line segments from (n, 2j) to − j, − − j to (n, 2j). We take the 2 2 alternating sum of terms along the path. This yields the coefficient of tn/2−j in the Alexander- Conway polynomial. If n is odd, we compute the alternating sum along the path (n, 2j + 1) n − 1 n − 1 to − j, − − j to (n, 2j + 1) to compute the coefficient of t(n−1)/2−j. 2 2 References [ADO92] Y. Akutsu, T. Deguchi, and T. Ohtsuki. Invariants of colored links. J. Knot Theory Ramifications, 1(2):161–184, 1992. 34 MATTHEW HARPER

1 2 −1 1 1 1 −2 1 6 6 2 −1 12 −3 0 −1 2 −1 26 −10 46 −14 1 −2 1 25 −12 1 1 0 1 −2 1 37 −26 6 85 −46 6 1 3 1 −2 1 10 14 1

31 41 51 52 61 1 2 −1 1 1 1 −2 1 12 −3 12 −3 0 −1 2 −1 35 −18 3 53 −24 5 1 0 1 −2 1 38 −37 18 −3 124 −71 24 −3 2 −1 0 −1 2 −1 25 −38 35 −12 1 169 −124 53 −12 1 1 −2 1 0 1 −2 1 37 −18 3 71 −24 3 1 0 1 −2 1 3 5 1 −2 1 1

62 63 71 6 6 26 −10 44 −14 13 39 −32 10 36 114 −68 18 64 −23 24 −37 32 −10 196 −68 164 −132 68 −14 97 −64 13 13 −24 39 −26 6 313 −196 −36 181 −164 114 −44 6 23 37 −32 10 68 132 −68 14 10 18

72 73 74 75 1 1 30 −5 30 −5 163 −60 7 209 −70 9 366 −215 60 −5 608 −301 70 −5 457 −366 163 −30 1 865 −608 209 −30 1 215 −60 5 301 −70 5 7 9

76 77

Figure 10. The value of ∆sl3 for all prime knots with fewer than seven crossings.

[BCGP16] C. Blanchet, F. Costantino, N. Geer, and B. Patureau-Mirand. Non semi-simple TQFTs, Reide- meister torsion and Kashaev’s invariants. Adv. Math., 301:1–78, 2016. [BN00] H. Boden and A. Nicas. Universal formulae for SU(n) Casson invariants of knots. Trans. Amer. Math. Soc., 352(7):3149–3187, 2000. [Bou02] N. Bourbaki. Lie Groups and Lie Algebras Chapters 4-6. Springer-Verlag, 2002. [Coc04] T. Cochran. Noncommutative knot theory. Algebr. Geom. Topol., 4:347–398, 2004. [CP95] V. Chari and A. Pressley. A Guide To Quantum Groups. Cambridge University Press, 1995. [CR21] T. Creutzig and M. Rupert. Uprolling unrolled quantum groups. Commun. Contemp. Math., doi: 10.1142/S0219199721500231. [DIL05] D. De Wit, A. Ishii, and J. Links. Inﬁnitely many two-variable generalisations of the Alexander- Conway polynomial. Algebr. Geom. Topol., 5(18):405–418, 2005. [FHL+85] P. Freyd, D. Yetter J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc., 12:239–246, 1985. A NON-ABELIAN GENERALIZATION OF THE ALEXANDER POLYNOMIAL FROM QUANTUM sl3 35

1 1 12 −3 20 −4 53 −24 5 132 −48 8 128 −83 36 −7 468 −236 76 −11 217 −164 101 −36 5 1108 −672 292 −76 8 344 −251 164 −83 24 −3 1924 −1328 672 −236 48 −4 433 −344 217 −128 53 −12 1 2353 −1924 1108 −468 132 −20 1 251 −164 83 −24 3 1328 −672 236 −48 4 101 −36 5 292 −76 8 7 11

89 817 1 1 42 −6 42 −6 420 −120 14 420 −120 14 1870 −802 186 −17 1972 −834 198 −19 4650 −2718 970 −186 14 5414 −2992 1046 −198 14 7296 −5446 2718 −802 120 −6 9680 −6582 2992 −834 120 −6 8257 −7296 4650 −1870 420 −42 1 11737 −9680 5414 −1972 420 −42 1 5446 −2718 802 −120 6 6582 −2992 834 −120 6 970 −186 14 1046 −198 14 17 19

932 933 1 12 −3 1 53 −24 5 2 −1 108 −79 36 −7 6 9 −4 1 105 −124 93 −36 5 46 −14 32 −13 4 −1 24 −91 124 −79 24 −3 85 −46 6 49 −32 9 −2 1 -23 −24 105 −108 53 −12 1 14 13 −4 1 91 −124 79 −24 3 1 93 −36 5 7

946 10132 10155 −4 −20 4 2 −4 8 8 −4 12 164 −46 12 −8 2 148 −34 476 −248 46 8 −4 0 496 −228 34 −12 649 −476 164 4 −20 4 0 721 −496 148 −12 -23 12 248 −46 −8 4 0 228 −34 12 −8 2 4 0 11n34 11n42 Wh (31)

Figure 11. The value of ∆sl3 for some prime knots with more than seven crossings.

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Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA Email address: [email protected]