aa r X i v : . [ m a t h . QA ] J u l THE ALEXANDER POLYNOMIAL AS A UNIVERSALINVARIANT
RINAT KASHAEV
Abstract.
Let B be the polynomial ring C [ a ± , b ] with the structure of acomplex Hopf algebra induced from its interpretation as the algebra of regu-lar functions on the affine linear algebraic group of complex invertible uppertriangular 2-by-2 matrices of the form (cid:0) a b (cid:1) . We prove that the universalinvariant of a long knot K associated to B is the reciprocal of the canonicallynormalised Alexander polynomial ∆ K ( a ). Given the fact that B admits a q -deformation B q which underlies the (coloured) Jones polynomials, our resultprovides another conceptual interpretation for the Melvin–Morton–Rozanskyconjecture proven by Bar-Nathan and Garoufalidis, and Garoufalidis and Lˆe. Introduction
The universal quantum knot invariants introduced and studied in a number ofworks [22, 14, 15, 20, 16, 3, 9, 26] is a convenient tool allowing to encode the multi-tude of quantum invariants associated to a given Hopf algebra into a single algebraicobject in completely representation independent way. As a result, the universalinvariants are of great potential for conceptual understanding and organisation ofthe diversity of quantum invariants though the increased computational complexitymakes them perhaps less useful for practical calculations. Nonetheless, as shows theexample of the logarithmic invariants of Murakami–Nagatomo [19], invariants asso-ciated to non semi-simple representations are sometimes better accessible throughthe universal invariants than directly from the R-matrix calculations.In this paper we address the problem of identification of the universal invariantof long knots in one of the simplest cases of non-trivial Hopf algebras, namely thecommutative but non co-commutative complex Hopf algebra B := C [ a ± , b ] withthe group-like element a and the element b with the co-product(1) ∆( b ) = a ⊗ b + b ⊗ . More abstractly, one can think of B as the algebra of regular functions C [Aff ( C )]on the affine linear algebraic group Aff ( C ) := G a ( C ) ⋊ G m ( C ) of invertible up-per triangular complex 2-by-2 matrices of the form ( a b ) where the Hopf algebrastructure is canonically induced by the group structure of Aff ( C ), see [27].The (maximal) Drinfeld’s quantum double of B is a Hopf algebra D ( B ) whichcontains two Hopf sub-algebras: B and the universal enveloping algebra of theLie algebra of Aff ( C ) generated by two primitive elements φ and ψ subject to thecommutation relation(2) φψ − ψφ = φ. Date : July 21, 2020.2020MSC: 57K16, 57K14, 57K10.Supported in part by the Swiss National Science Foundation, the subsidy no 200020 192081.
Within the algebra D ( B ), the element a is central while the element b interactswith φ and ψ through the commutation relations(3) φb − bφ = 1 − a, ψb − bψ = b. The formal universal R-matrix of D ( B )(4) R = (1 ⊗ a ) ψ ⊗ e φ ⊗ b = X m,n ≥ n ! (cid:18) ψm (cid:19) φ n ⊗ ( a − m b n , finds itself behind the associated universal invariant Z B ( K ) of a long knot K whichis a central element of a specific “profinite completion” of D ( B ) obtained as theconvolution algebra ( D ( B ) o ) ∗ of the co-algebra structure of the restricted dualHopf algebra D ( B ) o . The following main result of this paper was conjecturedin [10]. Theorem 1.
The universal invariant of a long knot K associated to the Hopfalgebra B is of the form Z B ( K ) = (∆ K ( a )) − where ∆ K ( t ) is the (canonicallynormalised) Alexander polynomial of K . The reciprocal of the Alexander polynomial in this theorem should be thoughtof as an element of the ring of formal power series C [[ a − D ( B ) where theelement a is always unipotent (that is a − Z [[ t − B can be q -deformed to a non-commutative Hopf algebra B q withthe same co-algebra structure (1) but with a q -commutative relation ab = qba . For q not a root of unity, the quantum double D ( B q ) contains the quantum group U q ( sl )as a Hopf sub-algebra. In particular, for each n ∈ Z > , it admits an n -dimensionalirreducible representation corresponding to the n -th coloured Jones polynomial. Inthe large n limit with q = t /n and fixed t , one recovers an infinite-dimensionalrepresentation of the Hopf algebra D ( B ) where the central element a is realisedby the scalar t . From that standpoint, Theorem 1 is consistent with the Melvin–Morton–Rozansky conjecture proven by Bar-Nathan and Garoufalidis in [1] and byGaroufalidis and Lˆe in [8].Theorem 1, in conjunction with the group-like nature of the element a , makesabsolutely transparent a result of Burau [4, 5] on the property of the Alexanderpolynomial related to cables: if one takes the n -th cable of a knot K and composesit with the braid of n strands which brings the first strand under all others to n -thposition, then the standard closure of the obtained string link gives a knot whoseAlexander polynomial is ∆ K ( t n ).The main tool of the proof of Theorem 1 is the use of a specific infinite dimen-sional representation of D ( B ) on a dense vector subspace A of a complex Hilbertspace of square integrable holomorphic functions on C considered over the algebraof formal power series C [[ ~ ]]. The evaluation of the formal universal R-matrix (4)under this representation is a well defined element of the algebra (End( A )) ⊗ [[ ~ ]],and it is this property of the R-matrix which, from one hand side, makes thecorresponding Reshetikhin–Turaev functor well defined as formal power series in ~ HE ALEXANDER POLYNOMIAL AS A UNIVERSAL INVARIANT 3 despite the infinite dimensionality of the representation, and from the other hand, itallows us to use the Gaussian integrals to express the quantum invariant in terms ofa minor of the unreduced Burau representation matrix. Similarly to the work [24],the identification of the quantum invariant with the Alexander polynomial is ac-complished through a direct relationship of the latter to a minor of the unreducedBurau representation matrix.
Theorem 2 ([13]) . Let a knot K be the closure of a braid β ∈ B n and ψ n ( β ) ∈ GL n ( Z [ t ± ]) the image of β under the unrestricted Burau representation (where theimages of the standard Artin generators are linear in t ). Let ˆ β n be the ( n − × ( n − matrix obtained from ψ n ( β ) by throwing away the n -th column and the n -th row.Then, the Alexander polynomial of K is given by the formula (5) ∆ K ( t ) = t − n − g ( β )2 det( I n − − ˆ β n ) where I k denotes the identity k × k matrix and g : B n → Z is the group homomor-phism that sends the Artin generators to 1. Notice that the exponent of t in the front factor of (5) is always an integer due toa specific parity property of the number g ( β ). A proof of Theorem 2, based on theAlexander–Conway skein relation, is outlined in [13]. As an independent proof, wedirectly relate (5) to another known determinantal formula for the Alexander poly-nomial [2, 12] that uses the reduced Burau representation and where the correctingmultiplicative factor is slightly more complicated. Outline.
Section 2 starts with a concise review of the definition of the universalinvariants from [10], and then describes the center of D ( B ). Remark that theuniversal invariant takes its values in a certain “profinite completion” of this cen-ter. Section 3 introduces few algebraic and analytic tools used in the subsequentsections: the Hilbert spaces H n of holomorphic functions on C n together with aparticular class of elements called Schr¨odinger’s coherent states (which are just lin-ear exponential functions), the dense subspaces A n ⊂ H n generated by products ofcoherent states and polynomials, and the Gaussian integration formula. The latteris the standard tool in quantum field theory which can also be thought of as an an-alytic version of MacMahon’s Master theorem [17]. In Section 4 the representationof D ( B ) in the space A [[ ~ ]] is introduced, the evaluation of the formal universalR-matrix is shown to be well defined and related to the basic building 2 × Acknowledgements.
I would like to thank Louis-Hadrien Robert and Rolandvan der Veen for useful discussions. This work is partially supported by the SwissNational Science Foundation, the subsidy no 200020 192081.2.
Universal invariants of long knots from Hopf algebras
In this section, based on the construction of R-matrix invariants of long knotsin [23, 22, 11], we briefly describe the definition of the universal invariants of longknots given in [10], see also [16, 26] for an approach through the co-end.
RINAT KASHAEV
Consider the category
Hopf K of Hopf algebras over a field K with invertibleantipode. The restricted dual of an algebra provides us with a contravariant end-ofunctor ( · ) o : Hopf K → Hopf K which associates to a Hopf algebra H with multi-plication ∇ the Hopf algebra(6) H o := ( ∇ ∗ ) − ( H ∗ ⊗ H ∗ ) ⊂ H ∗ whose underlying vector space is the vector subspace of the algebraic dual H ∗ generated by all matrix coefficients of all finite dimensional representations of H [7].Drinfeld’s quantum double of H ∈ Ob Hopf K (see, for example, [18]) is a Hopfalgebra D ( H ) ∈ Ob Hopf K uniquely determined by the property that there aretwo Hopf algebra inclusions(7) ı : H → D ( H ) , : H o, op → D ( H )such that D ( H ) is generated by their images subject to the commutation relations(8) ( f ) ı ( x ) = h f (1) , x (1) ih f (3) , S ( x (3) ) i ı ( x (2) ) ( f (2) ) ∀ ( x, f ) ∈ H × H o where we use Sweedler’s notation for the co-multiplication(9) ∆( x ) = x (1) ⊗ x (2) , (∆ ⊗ id)(∆( x )) = x (1) ⊗ x (2) ⊗ x (3) , . . . The restricted dual of the quantum double D ( H ) o is a dual quasi-triangular Hopfalgebra with the dual universal R-matrix(10) ̺ : D ( H ) o ⊗ D ( H ) o → K , x ⊗ y
7→ h x, ( ı o ( y )) i which, among other things, satisfies the Yang–Baxter relation(11) ̺ , ∗ ̺ , ∗ ̺ , = ̺ , ∗ ̺ , ∗ ̺ , in the convolution algebra (( D ( H ) o ) ⊗ ) ∗ . If { e i } i ∈ I is a linear basis of H and { e i } i ∈ I is the associated set of canonical (dual) linear forms, then one can write aformal universal R-matrix(12) R := X i ∈ I ( e i ) ⊗ ı ( e i )as the formal conjugate of the dual universal R-matrix in the sense of the equality(13) h x ⊗ y, R i = h ̺, x ⊗ y i ∀ x, y ∈ D ( H ) o . Furthermore, for any finite-dimensional right co-module(14) V → V ⊗ D ( H ) o , v v (0) ⊗ v (1) , the dual universal R-matrix gives rise to a rigid R-matrix(15) r V : V ⊗ V → V ⊗ V, u ⊗ v v (0) ⊗ u (0) h ̺, v (1) ⊗ u (1) i . This implies that there exists a universal invariant of long knots Z H ( K ) taking itsvalues in the center of the convolution algebra ( D ( H ) o ) ∗ such that(16) J r V ( K ) v = v (0) h Z H ( K ) , v (1) i ∀ v ∈ V where J r V ( K ) ∈ End( V ) is the invariant of long knots associated to r V , see [10] fordetails. HE ALEXANDER POLYNOMIAL AS A UNIVERSAL INVARIANT 5
The Hopf algebra D ( B ) and its center. Recall that B is the polynomialalgebra C [ a ± , b ] provided with the structure of a Hopf algebra where a is a group-like element and the co-product of b is given in (1).The opposite B o, op1 of the restricted dual Hopf algebra B o is composed of twoHopf sub-algebras: the group algebra C [Aff ( C )] generated by group-like elements(17) χ u,v , ( u, v ) ∈ C × C =0 , χ u,v χ u ′ ,v ′ = χ u + vu ′ ,vv ′ , and the universal enveloping algebra U (Lie Aff ( C )) generated by two primitiveelements ψ and φ satisfying the relation (2). The relations between the generatorsof C [Aff ( C )] and U (Lie Aff ( C )) are of the form(18) [ χ u,v , ψ ] = uφχ u,v , χ u,v φ = vφχ u,v ∀ ( u, v ) ∈ C × C =0 where [ x, y ] := xy − yx . As linear forms on B , they are defined by the relations(19) h χ u,v , b m a n i = u m v − m − n , h φ, b m a n i = δ m, , h ψ, b m a n i = δ m, n, ∀ ( m, n ) ∈ Z ≥ × Z . The commutation relations (8) in the case of the quantum double D ( B ) takethe form(20) [ ψ, b ] = b, [ φ, b ] = 1 − a, bχ u,v = χ u,v ( bv + ( a − u ) ∀ ( u, v ) ∈ C × C =0 and a is central. The formal universal R-matrix is given by formula (4).Any finite dimensional right co-module V over D ( B ) o is canonically a left mod-ule over D ( B ) defined by(21) xw = w (0) h w (1) , x i , ∀ ( x, w ) ∈ D ( B ) × V, where the elements a − b and φ are necessarily nilpotent, so that the formalinfinite double sum in (4) truncates to a well defined finite sum. Lemma 1.
The center of the algebra D ( B ) is the polynomial sub-algebra C [ a ± , c ] where (22) c := φb + ( a − ψ. Proof.
It is easily verified that c is central. Any element x ∈ D ( B ) can uniquelybe written in the form(23) x = X ( u,v,m ) ∈ C × C =0 × Z χ u,v e m p u,v,m ( a, c, ψ ) , where(24) e m := b m if m > m = 0; φ − m if m < p u,v,m ( a, c, ψ ) ∈ C [ a ± , c, ψ ] is non-zero for only finitely many triples ( u, v, m ).Assume that x is central. Then, for any s ∈ C =0 , we have the equality(25) x = χ − ,s xχ ,s = X ( u,v,m ) ∈ C × C =0 × Z χ u/s,v e m s m p u,v,m ( a, c, ψ )= X ( u,v,m ) ∈ C × C =0 × Z χ u,v e m s m p us,v,m ( a, c, ψ ) RINAT KASHAEV which implies that for any fixed triple ( u, v, m ) ∈ C × C =0 × Z , one has the familyof equalities(26) p u,v,m = s m p us,v,m ∀ s ∈ C =0 . This means that p u,v,m can only be non-zero if u = m = 0. Thus, the element x takes the form(27) x = X v ∈ C =0 χ ,v p ,v, ( a, c, ψ ) . The equality(28) bx = xb = b X v ∈ C =0 χ ,v v − p ,v, ( a, c, ψ + 1) . is equivalent to the equalities(29) p ,v, ( a, c, ψ + 1) = v − p ,v, ( a, c, ψ ) ∀ v ∈ C =0 which imply that the polynomial p ,v, ( a, c, ψ ) can be non-zero only if v = 1 and ifit does not depend on ψ . We conclude that x ∈ C [ a ± , c ]. (cid:3) Schr¨odinger’s coherent states
Here we briefly review the theory of standard Schr¨odinger’s coherent states (seefor example [21]).For any n ∈ Z > , let H n ⊂ L ( C n , µ n ) be the complex Hilbert space of squareintegrable holomorphic functions f : C n → C with the scalar product(30) h f | g i := Z C n f ( z ) g ( z ) d µ n ( z )where the measure µ n on C n is absolutely continuous with respect to the Lebesguemeasure λ n on C n ≃ R n with the Radon–Nikodym derivative(31) d µ n d λ n ( z ) = 1 π n e −k z k , k z k := rX n − i =0 | z i | . By direct calculation, one verifies that the monomials(32) e k ( z ) := n − Y i =0 z k i i √ k i ! , k ∈ Z n ≥ form an orthonormal family in H n which is, in fact, a Hilbert basis due to thevalidity of Taylor’s (multivariable) expansion for holomorphic functions:(33) f ( z ) = X k ∈ Z n ≥ n − Y i =0 z k i i k i ! ∂ k i f ( w ) ∂w k i i (cid:12)(cid:12)(cid:12)(cid:12) w =0 = X k ∈ Z n ≥ e k ( z ) n − Y i =0 √ k i ! ∂ k i f ( w ) ∂w k i i (cid:12)(cid:12)(cid:12)(cid:12) w =0 which, in the case when f ∈ H n , implies that(34) Z C n e k ( z ) f ( z ) d µ n ( z ) = n − Y i =0 √ k i ! ∂ k i f ( w ) ∂w k i i (cid:12)(cid:12)(cid:12)(cid:12) w =0 ∀ k ∈ Z n ≥ . For any u ∈ C n , multiplying both sides of (34) by e k ( u ), summing over all k ∈ Z n ≥ and, using the Fubini (or dominant convergence) theorem in the left hand side for HE ALEXANDER POLYNOMIAL AS A UNIVERSAL INVARIANT 7 exchanging the integration and summation, and the Taylor formula (33) in the righthand side, we obtain(35) Z C n ϕ u (¯ z ) f ( z ) d µ n ( z ) = f ( u ) ∀ f ∈ H n where the holomorphic function(36) ϕ u : C n → C , z X k ∈ Z n ≥ e k ( u ) e k ( z ) = e P n − i =0 u i z i determines an element ϕ u ∈ H n called (Schr¨odinger’s) coherent state . By treatingelements of C n as column vectors we can write ϕ u ( z ) = e u ⊤ z . Let us also adopt thenotation w ∗ := ¯ w ⊤ for the Hermitian conjugation, i.e. the transposition combinedwith the complex conjugation. With this notation we have the equalities(37) k w k = w ∗ w, ϕ u (¯ z ) = e z ∗ u . The integral formula (35) expresses the reproducing property of coherent states(38) h ϕ ¯ u | f i = f ( u ) ∀ ( f, u ) ∈ H n × C n . The choice f = ϕ v in the last formula gives the scalar product between the coherentstates(39) h ϕ ¯ u | ϕ v i = ϕ v ( u ) = ϕ u ( v ) = e u ⊤ v . In particular, the norm of a coherent state ϕ v is determined by the Euclidean normof v through the formula(40) k ϕ v k = e k v k / . A dense subspace of H n . Another useful property of the coherent states isthat the (dense) vector subspace A n of H n generated by products of coherent statesand polynomials is stable under the multiplication of elements of A n as functionsso that A n carries the additional structure of a commutative algebra, and it is inthe domain of any linear differential operator with coefficients in A n . For example,when n = 1, the Hilbert basis of H given by the monomials { e k } k ∈ Z ≥ ⊂ A isthe eigenvector basis of the 1-dimensional quantum harmonic oscillator with the(self-adjoint) Hamiltonian operator z ∂∂z .3.2. Gaussian integration formula.
Writing out explicitly the scalar product asan integral in (39), we obtain an integral identity(41) Z C n e v ⊤ z + z ∗ u d µ n ( z ) = e v ⊤ u which is a special case of the general Gaussian integration formula(42) Z C n e v ∗ z + z ∗ u + z ∗ Mz d µ n ( z ) = e v ∗ W − u det( W ) , W := I n − M where M is an arbitrary complex n -by- n matrix sufficiently close to zero so thatthe integral is absolutely convergent. Furthermore, the expansion of (42) in powerseries in M with u = v = 0 corresponds to the purely combinatorial MacMahonMaster theorem [17]. RINAT KASHAEV Representations of D ( B ) in A [[ ~ ]]Recall that A is the vector subspace of H generated by products of coherentstates with polynomials. For any λ ∈ C , the mappings(43) a ~ , b ∂∂z , φ ~ z, ψ λ − z ∂∂z and the action(44) χ u,v f ( z ) = e ~ uz f ( vz )determine a homomorphism of algebras(45) ρ λ : D ( B ) → End( A [[ ~ ]])which sends the central element c defined in (22) to λ ~ .An important property of the representation ρ λ is that the image under ρ ⊗ λ ofthe formal R-matrix (4) is a well defined element of the algebra End( A ) ⊗ [[ ~ ]]:(46) ρ ⊗ λ ( R ) = (1 + ~ ) λ − z ∂∂z e ~ z ∂∂z = X m,n ≥ ~ m + n n ! (cid:18) λ − z ∂∂z m (cid:19)(cid:16) z ∂∂z (cid:17) n . In particular, the double sum in (46) truncates to a finite sum if the indeterminate ~ is nilpotent. Thus, despite the fact that the representation ρ λ is infinite dimensional,the corresponding R-matrix is well suited for calculation of the image under ρ λ ofthe universal invariant Z B ( K ). Moreover, as the parameter λ enters only throughthe overall normalisation factor (1 + ~ ) λ of the R-matrix, the associated invariantis independent of λ . For that reason, in what follows, we put λ = 0 and work onlywith the representation ρ := ρ .In order to apply the construction of [10], we define the input R-matrix(47) r := ρ ⊗ ( R ) P where P ∈ Aut( A ) is the permutation operator acting by exchanging the argu-ments. By using (46), we obtain the following explicit action of r :(48) rf ( z ) = rf ( z , z ) = (1 + ~ ) − z ∂∂z f ( z + ~ z , z )= f (cid:16) z + ~ ~ z ,
11 + ~ z (cid:17) = f ( U ⊤ z )where(49) U := (cid:18) ~ ~ ~ (cid:19) = (cid:18) − t t (cid:19) , t := 11 + ~ , is the 2-by-2 matrix entering the definition of the (unrestricted) Burau representa-tion of the braid groups [6]. The action of r on the coherent states is realized bythe action of the transposed matrix on the space of parameters:(50) rϕ v ( z ) = ϕ v ( U ⊤ z ) = ϕ Uv ( z ) . In what follows, we use the indeterminate t defined in terms of ~ through theformula in (49). HE ALEXANDER POLYNOMIAL AS A UNIVERSAL INVARIANT 9
The diagrammatic rules for the Reshetikhin–Turaev functor.
Fromthe formula (50), one calculates the integral kernel of r with respect to the coherentstates(51) h ϕ w | r | ϕ v i = h ϕ w ,w | r | ϕ v ,v i = e w ∗ Uv which corresponds to the value of the Reshetikhin–Turaev functor associated to pos-itive crossings of all orientations in normal long knot diagrams with edges colouredby complex numbers:(52) v v w w , w v w v , w w v v , v w v w RT r ϕ w ,w | r | ϕ v ,v i Likewise, the integral kernel of r − given by the formula(53) h ϕ w | r − | ϕ v i = h ϕ w ,w | r − | ϕ v ,v i = e w ∗ U − v is associated to negative crossings of all orientations:(54) v v w w , w v w v , w w v v , v w v w RT r ϕ w ,w | r − | ϕ v ,v i . We complete the list of the diagrammatic rules by adding the rules for verticalsegments and local extrema(55) wv , vw , w v , w v RT r e ¯ wv where e ¯ wv is the integral kernel of the identity operator id A :(56) h ϕ w | id A | ϕ v i = h ϕ w | ϕ v i = e ¯ wv . For later use, we calculate the following two Reshetikhin–Turaev images(57) h ϕ w | RT r (cid:16) (cid:17) | ϕ v i = RT r (cid:16) v w (cid:17) = Z C h ϕ w,u | r | ϕ v,u i d µ ( u )= Z C e ( ¯ w ¯ u ) (cid:16) − t t (cid:17) ( vu ) d µ ( u ) = Z C e ¯ w (1 − t ) v + ¯ wtu +¯ uv d µ ( u ) = e ¯ wv and(58) h ϕ w | RT r (cid:16) (cid:17) | ϕ v i = RT r (cid:16) v w (cid:17) = Z C h ϕ w,u | r − | ϕ v,u i d µ ( u )= Z C e ( ¯ w ¯ u ) (cid:16) t − − t − (cid:17) ( vu ) d µ ( u ) = Z C e ¯ wu +¯ ut − v +¯ u (1 − t − ) u d µ ( u ) = te ¯ wv where the integrals are calculated by using the Gaussian integration formula (42). Proof of Theorem 1
Let K be represented by the closure of a braid β ∈ B n . Let us choose a normallong knot diagram D β representing K according to the picture(59) D β = D β = β . . .. . .. . . with the writhe g ( D β ) = g ( β )+ n − ψ n ( β ) in the block form(60) ψ n ( β ) = (cid:18) ˆ β n b β c β d β (cid:19) , and using the general Gaussian integration formula (42), we calculate(61) h ϕ w | RT r ( D β ) | ϕ v i = RT r (cid:18) D β wv (cid:19) = Z C n − e ( u ∗ ¯ w ) ψ n ( β ) ( uv ) d µ n − ( u )= Z C n − e ( u ∗ ¯ w ) (cid:18) ˆ β n b β c β d β (cid:19) ( uv ) d µ n − ( u )= Z C n − e ¯ wd β v + ¯ wc β u + u ∗ b β v + u ∗ ˆ β n u d µ n − ( u ) = e ¯ wd β v + ¯ wc β ( I n − − ˆ β n ) − b β v det( I n − − ˆ β n ) . On the other hand, given the fact that we are calculating a central element realisedby a scalar so that on `a priori grounds the result should be proportional to theintegral kernel of the identity operator e ¯ wv , we conclude that the identity(62) d β + c β ( I n − − ˆ β n ) − b β = 1is satisfied, a property of ψ n ( β ) which does not look to be easy to prove withoutpassing through the Gaussian integration and referring to the universal invariant.Finally, it remains to take into account the writhe correction, which, accordingto the values in (57) and (58) is given by the formula(63) h ϕ w | RT r (cid:0) ξ − g ( D β ) / (cid:1) | ϕ v i e − ¯ wv = t g ( D β ) / = t ( g ( β )+ n − / where we use the notation ξ k from [10] for a specific class of long knot diagramsused to compensate the writhe of the diagram. Putting together (61) and (63), theresult for the invariant J r ( K ) reads(64) h ϕ w | J r ( K ) | ϕ v i e − ¯ wv = t ( g ( β )+ n − / det( I n − − ˆ β n ) = 1∆ K ( t )where the last equality is due to formula (5). Taking into account the relationbetween ~ , t and the realisation of the central element a of D ( B ) as well as thesymmetry of the Alexander polynomial under the substitution t t − , we concludethe proof. HE ALEXANDER POLYNOMIAL AS A UNIVERSAL INVARIANT 11 Proof of Theorem 2
In this section, we adopt the notation of [12] and first briefly describe the unre-duced and reduced Burau representations of the braid groups B n for n ≥ k ≥
1, denote by I k the identity k × k matrix. Let(65) ψ n : B n → GL n (Λ) , Λ := Z [ t ± ] , be the unrestricted Burau representation where Artin’s standard generators σ i ,1 ≤ i < n , are realised by the matrices(66) ψ n ( σ i ) = U i := I i − ⊕ U ⊕ I n − i − . For any k ≥
1, define the invertible upper triangular k × k matrix(67) C k = X ≤ i ≤ j ≤ k E i,j = I k + X ≤ i 1, of thematrix ψ rn ( β ) − I n − through the formula (73) (1 − t n ) ∗ β = n − X i =1 ( t i − a i . Lemma 2. Let ˆ β n be the ( n − × ( n − matrix obtained from ψ n ( β ) by throwingaway the n -th row and the n -th column. Then, one has the following equality in Λ : (74) ( t − n − 1) det( ˆ β n − I n − ) = ( t − − 1) det( ψ rn ( β ) − I n − ) . This is the content of Lemma 3.10 of [12] where the formula is written with a typo. Proof. We have the following equality of matrices:(75) ˆ β n = ( C n − ψ rn ( β ) + 1 n − ∗ β ) C − n − ⇔ C − n − ˆ β n C n − = ψ rn ( β ) + C − n − n − ∗ β = ψ rn ( β ) + (cid:0) n − (cid:1) ∗ β . One can verify this by explicit calculation based on the block structure (70):(76) ψ n ( β ) = (cid:18) C n − n − ⊤ n − (cid:19) (cid:18) ψ rn ( β ) 0 n − ∗ β (cid:19) C − n = (cid:18) C n − ψ rn ( β ) + 1 n − ∗ β ∗∗ β (cid:19) (cid:18) C − n − ∗ ⊤ n − (cid:19) = (cid:18) ( C n − ψ rn ( β ) + 1 n − ∗ β ) C − n − ∗∗ (cid:19) . Thus,(77) det( ˆ β n − I n − ) = det (cid:0) ψ rn ( β ) − I n − + (cid:0) n − (cid:1) ∗ β (cid:1) = det a ... a n − a n − + ∗ β . By multiplying both sides of (77) by (1 − t n ) and using (73), we obtain(1 − t n ) det( ˆ β n − I n − ) = det a ... a n − (1 − t n )( a n − + ∗ β ) = det a ... a n − (1 − t n ) a n − + P n − i =1 ( t i − a i = det a ... a n − ( t − − t n a n − = ( t − − t n det( ψ rn ( β ) − I n − )where in the third equality we dropped from the sum all the terms proportional tothe rows different from n − (cid:3) Proof of Theorem 2. 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