aa r X i v : . [ m a t h . QA ] D ec INVARIANTS OF LONG KNOTS
RINAT KASHAEV
Dedicated to Nikolai Reshetikhin on the occasion of his 60th birthday
Abstract.
By using the notion of a rigid R-matrix in a monoidal categoryand the Reshetikhin–Turaev functor on the category of tangles, we reviewthe definition of the associated invariant of long knots. In the frameworkof the monoidal categories of relations and spans over sets, by introducingracks associated with pointed groups, we illustrate the construction and theimportance of consideration of long knots. Else, by using the restricted dualof algebras and Drinfeld’s quantum double construction, we show that to anyHopf algebra H with invertible antipode, one can associate a universal longknot invariant Z H ( K ) taking its values in the convolution algebra (( D ( H )) o ) ∗ of the restricted dual Hopf algebra ( D ( H )) o of the quantum double D ( H ) of H . That extends the known constructions of universal invariants previouslyconsidered mostly either in the case of finite dimensional Hopf algebras or byusing some topological completions. Introduction
This paper is to a large extent a review of certain aspects of quantum invariantswhere we restrict ourselves exclusively to the context of long knots. More generally,one can consider also the string links, but it is known that the topological classesof string links are in bijection with the classes of ordinary links only if the numberof components is one, i.e. if a string link is a long knot.We describe in detail the construction of invariants of long knots by using rigidR-matrices (solutions of the quantum Yang–Baxter relation) in monoidal categories.The importance of long knots (as opposed to usual closed knots) is illustrated byconsidering a general class of group-theoretical R-matrices put into the context ofmonoidal categories of relations and spans over sets. These R-matrices are indexedby pointed groups that is groups with a distinguished element. The underlyingracks seem not to be considered previously in the existing literature.Drinfeld’s quantum double construction gives rise to a large class of rigid R-matrices, and the associated invariants get factorized through universal invariantsassociated to underlying Hopf algebras. Such universal invariants were introducedand studied in a number of works [15, 9, 10, 14, 11, 2, 6, 18] mostly either inthe context of finite dimensional Hopf algebras or certain topological completions,for example by considering formal power series. Here, we define the universalinvariants purely algebraically and with minimal assumptions on the underlyingHopf algebras. In particular, we emphasize the case of infinite dimensional Hopfalgebras. The distinguishing feature of our approach is the use of the restricted orfinite dual of an algebra in conjunction with the quantum double construction.
Date : December 31, 2019.
The outline of the paper is as follows.In section 1 we recall the definitions of long knots and their diagrams and intro-duce the notions of a normal diagram and a normalization of an arbitrary diagram.In section 2 we recall the definition of a rigid R-matrix in a monoidal categoryand give a detailed description of a long knot invariant associated to a given rigidR-matrix.In section 3 we consider a special class of rigid R-matrices in the categories ofrelations and spans over sets. Each such R-matrix is associated to a pointed groupwith a canonical structure of a rack, and Theorem 2 identifies the associated invari-ant with the set of representations of the knot group into the group that underliesthe rack. The example of an extended Heisenberg group gives rise to an invariantideal in the polynomial algebra Q [ t, t − , s ] closely related but not equivalent to theAlexander polynomial, at least if the latter admits higher multiplicity roots.In section 4, based on the restricted dual of an algebra and Drinfeld’s quan-tum double construction, we describe a universal invariant associated to any Hopfalgebra with invertible antipode. Acknowledgements.
I would like to thank Bruce Bartlett, L´eo B´enard, JoanPorti, Louis-HadrienRobert, Arkady Vaintrob, Roland van der Veen and AlexisVirelizier for valuable discussions.1.
Long knots
Definition 1.
An embedding f : R → R is called long knot if there exist a, b ∈ R such that f ( t ) = (0 , , t ) for any t < a or t > b .Two long knots f, g : R → R are called equivalent if they are ambient isotopic,that is if there exists an ambient isotopy(1) H : R × [0 , → R × [0 , , H ( x, t ) = ( h t ( x ) , t ) , h = id R , such that, for any t ∈ [0 , h t ◦ f is a long knot and g = h ◦ f .A long knot is called tame or regular if it is equivalent to a polygonal long knot.An (oriented) long knot diagram D is a (1,1)-tangle diagram in R representinga tame long knot(2) D = D .
Two long knot diagrams are called (Reidemeister) equivalent if they can be re-lated to each other by a finite sequence of oriented Reidemeister moves of all types.
Remark 1.
The set of long knot diagrams is a monoid with respect to the com-position(3) D ◦ D ′ := DD ′ Remark 2.
In the case of long knots, the Reidemeister theorem states that twolong knot diagrams are equivalent if and only if the corresponding long knots are
NVARIANTS OF LONG KNOTS 3 equivalent. Furthermore, a folklore theorem states that the natural map of longknot diagrams to closed oriented knot diagrams(4) D D induces a bijection between the respective Reidemeister equivalence classes. Inparticular, any invariant of long knots is also an invariant of closed knots.In what follows, we will always assume that a long knot diagram is put intoa generic position with respect to the vertical axis so that all crossing have non-vertical strands as in the letter X. We denote by w ( D ) the writhe of D defined asthe number of positive crossings minus the number of negative crossings where acrossing is called positive if the ordered pair of tangent vectors ( v overpass , v underpass )induces the standard orientation of R . Definition 2.
A (long knot) diagram is called normal if it has no local extrema(with respect to vertical direction) oriented from left to right like and .To any diagram D , we associate its normalization ˙ D , the diagram obtained from D by the replacements(5) , . It will be of special interest for us the normal long knot diagrams(6) ξ + := , ξ − := , ξ n := if n = 0; ξ sgn( n ) ◦ · · · ◦ ξ sgn( n ) | {z } | n | times if n = 0where n ∈ Z , sgn( n ) := n/ | n | and we identify the signs ± with the numbers ± Remark 3.
Any normal long knot diagram has an even number of crossings. Inparticular, we have w ( ξ n ) = 2 n .2. Invariants of long knots from rigid R -matrices We say an object G of a monoidal category C (with tensor product ⊗ and unitobject I ) admits a left adjoint if there exists an object F and morphisms(7) ε : F ⊗ G → I , η : I → G ⊗ F such that(8) ( ε ⊗ id F ) ◦ (id F ⊗ η ) = id F , (id G ⊗ ε ) ◦ ( η ⊗ id G ) = id G . In that case, the quadruple (
F, G, ε, η ) is called adjunction in C . Definition 3.
Let C be a monoidal category. An R-matrix over an object G ∈ Ob C is an element r ∈ Aut( G ⊗ G ) that satisfies the Yang–Baxter relation(9) ( r ⊗ id G ) ◦ (id G ⊗ r ) ◦ ( r ⊗ id G ) = (id G ⊗ r ) ◦ ( r ⊗ id G ) ◦ (id G ⊗ r ) . Definition 4.
Let (
F, G, ε, η ) be an adjunction in a monoidal category C . An R -matrix r over G is called rigid if the morphisms(10) g r ± := ( ε ⊗ id G ⊗ F ) ◦ (id F ⊗ r ± ⊗ id F ) ◦ (id F ⊗ G ⊗ η )are invertible. RINAT KASHAEV
We also denote(11) gg r ± := ( ε ⊗ id F ⊗ F ) ◦ (id F ⊗ g r ± ⊗ id F ) ◦ (id F ⊗ F ⊗ η ) . One easily checks the identity(12) gg r − = (cid:16)ee r (cid:17) − . Associated to a rigid R-matrix r over G with an adjunction ( F, G, ε, η ), the
Reshetikhin–Turaev functor RT r associates to any normal long knot diagram D theendomorphism RT r ( D ) : G → G obtained as follows.Assuming that the non-trivial part of D is contained in R × [0 , t < t < · · · < t n − < t n = 1 such that, for any i ∈ { , . . . , n − } , the intersection D i := D ∩ ( R × [ t i , t i +1 ]) is an ordered (from leftto right) finite sequence of connected components each of which is isotopic relativeto boundary either to one of the four types of segments(13) , , , or to one of the eight types of crossings(14) , , , , , , , . To such an intersection, we associate a morphism f i in C by taking the tensorproduct (from left to right) of the morphisms associated to the connected fragmentsof D i according to the following rules:(15) id G , id F , ε, η, (16) r, r − , (17) e r, g r − , (18) ee r, gg r − , (19) (cid:16)g r − (cid:17) − , ( e r ) − . The morphism RT r ( D ) : G → G associated to D is obtained as the composition(20) RT r ( D ) := f n − ◦ · · · ◦ f ◦ f . Example 1.
Let D = ξ − defined in (6). Then RT r ( D ) = f ◦ f ◦ f ◦ f where(21) f = id G ⊗ η, f = id G ⊗ ( e r ) − , f = ( e r ) − ⊗ id G , f = ε ⊗ id G . Theorem 1 ([16, 15, 8]) . Let r be a rigid R -matrix over an object G of a monoidalcategory C with an adjunction ( F, G, ε, η ) . Then, for any long knot diagram D , theelement (22) J r ( D ) := RT r ( ˙ D ◦ ξ − w ( ˙ D ) / ) ∈ End( G ) depends on only the Reidemeister equivalence class of D . NVARIANTS OF LONG KNOTS 5
Proof.
Let ˙ ∼ be the equivalence relation on the set of normal long knot diagramsgenerated by the oriented versions of the Reidemeister moves RII and RIII and themoves R0 ± defined by the pictures(23) R0 + ←→ , R0 − ←→ with two possible orientations for the straight segment and two possibilities for thecrossing. The strategy of the proof is to show first the implication(24) ˙ D ˙ ∼ ˙ D ′ ⇒ RT r ( ˙ D ) = RT r ( ˙ D ′ )which, by taking into account the implication(25) ˙ D ˙ ∼ ˙ D ′ ⇒ w ( ˙ D ) = w ( ˙ D ′ ) , ensures the invariance of J r ( D ) under all Reidemeister moves RII and RIII, andthen to verify the invariance under the Reidemeister moves RI. It is in this last partof the proof where the correction of ˙ D by ξ − w ( ˙ D ) / is crucial.Invariance of Reshetikhin–Turaev functor with respect to moves R0 ± followsfrom the equivalences(26) ˙ ∼ ⇔ ˙ ∼ ⇔ ˙ ∼ , (27) ˙ ∼ ⇔ ˙ ∼ ⇔ ˙ ∼ and the definitions (10) and (11) of g r ± and gg r ± .Invariance of RT r with respect to oriented RII moves are easily checked first foreight basic moves(28) ˙ ∼ ˙ ∼ RT r r − ◦ r = id G ⊗ G = r ◦ r − , (29) ˙ ∼ ˙ ∼ RT r e r ◦ ( e r ) − = id G ⊗ F = g r − ◦ (cid:16)g r − (cid:17) − , (30) ˙ ∼ ˙ ∼ RT r (cid:16)g r − (cid:17) − ◦ g r − = id F ⊗ G = ( e r ) − ◦ e r, (31) ˙ ∼ ˙ ∼ RT r gg r − ◦ ee r = id F ⊗ F = ee r ◦ gg r − , and then for two composite moves(32) ˙ ∼ ˙ ∼ ˙ ∼ =with two possible choices for the crossings. RINAT KASHAEV
In order to check invariance of RT r with respect to RIII moves, we remark thataltogether there are 48 such moves which can indexed by the set Sym(3) × {± } as follows.Given an RIII move, we enumerate the strands that intervene the move byfollowing their bottom open ends from left to right(33) ˙ ∼ ˙ ∼ and we define the associated element ( σ, ε ) ∈ Sym(3) × {± } by the conditionsthat for any i ∈ { , , } , σ ( i ) is the number of arcs on the i -th strand and ε i = 1if i -th strand is oriented upwards. For example, the RIII move(34) ˙ ∼ corresponds to permutation σ = (2 ,
3) = (1)(2 ,
3) and ε = (1 , − ,
1) while thepair ( σ = id , ε = (1 , , ∼ RT r r ◦ r ◦ r = r ◦ r ◦ r with the notations r := r ⊗ id G and r := id G ⊗ r .One can now show that by using the moves RII and R0 ± , any RIII move isequivalent to the reference move (35).Indeed, in the case ε = −
1, we have the equivalences(36) ˙ ∼ ⇔ ˙ ∼ ⇔ ˙ ∼ which imply the equivalence(37) ( σ, ( − , ε , ε )) ⇔ ( σ ◦ (1 , , , ( ε , ε , × {± } thus allowing to reduce the number of negative compo-nents of ε . Additionally, a right action of the permutation group Sym(3) on the setSym(3) × {± } is induced by the equivalences(38) ˙ ∼ ⇔ ˙ ∼ ⇔ ˙ ∼ and(39) ˙ ∼ ⇔ ˙ ∼ ⇔ ˙ ∼ which correspond to the respective equivalences(40) ( σ, ε ) ⇔ ( σ, ε ) ◦ (1 ,
2) and ( σ, ε ) ⇔ ( σ, ε ) ◦ (2 , × {± } where we interpret ( σ, ε ) ∈ Sym(3) × {± } as the map(41) ( σ, ε ) : { , , } → { , , } × {± } , i ( σ ( i ) , ε i ) . Thus, in conjunction with the equivalence (37), the right action of the group Sym(3)on the set Sym(3) × {± } establishes the equivalence of any RIII move to the NVARIANTS OF LONG KNOTS 7 reference move (35) and thereby the invariance of RT r with respect to all RIIImoves.Finally, in order to prove invariance of J r with respect to all RI moves, we needto check only the invariance with respect four basic moves of the form(42) ∼ as all others are consequences of the basic ones and the intermediate equivalencerelation ˙ ∼ generated by the moves R0 ± , RII and RIII:(43) = ∼ ˙ ∼ ˙ ∼ ⇒ ∼ and(44) = ∼ ˙ ∼ . Les us analyse the four cases of (42) separately.Case 1. If diagrams D and D ′ differ by the fragments(45) D ∋ , ∈ D ′ , then, by the definition of the normalisation of a long knot diagram, we have theequality ˙ D = ˙ D ′ . Thus, J r ( D ) = J r ( D ′ ).Case 2. Diagrams D and D ′ differ by the fragments(46) D ∋ , ∈ D ′ so that the normalised diagrams ˙ D and ˙ D ′ differ by the fragments(47) ˙ D ∋ , ∈ ˙ D ′ which imply that(48) w ( ˙ D ) = 2 + w ( ˙ D ′ ) ⇒ ξ + ◦ ξ − w ( ˙ D ) / = ξ − w ( ˙ D ′ ) / . On the other hand, we have the equivalence(49) ˙ D ∋ ˙ ∼ ∈ ˙ D ′ ◦ ξ + which, together with (48), implies that(50) ˙ D ◦ ξ − w ( ˙ D ) / ˙ ∼ ˙ D ′ ◦ ξ + ◦ ξ − w ( ˙ D ) / = ˙ D ′ ◦ ξ − w ( ˙ D ′ ) / ⇒ J r ( D ) = J r ( D ′ ) . Case 3. Diagrams D and D ′ differ by the fragments(51) D ∋ , ∈ D ′ so that(52) ˙ D ∋ ˙ ∼ ∈ ˙ D ′ ⇒ J r ( D ) = J r ( D ′ ) . Case 4. Diagrams D and D ′ differ by the fragments(53) D ∋ , ∈ D ′ so that we have for the corresponding normalised diagrams(54) ˙ D ∋ ˙ ∼ ∈ ˙ D ′ ◦ ξ − RINAT KASHAEV and(55) w ( ˙ D ) = w ( ˙ D ′ ) − ⇒ ξ − ◦ ξ − w ( ˙ D ) / = ξ − w ( ˙ D ′ ) / . Thus,(56) ˙ D ◦ ξ − w ( ˙ D ) / ˙ ∼ ˙ D ′ ◦ ξ − ◦ ξ − w ( ˙ D ) / = ˙ D ′ ◦ ξ − w ( ˙ D ′ ) / ⇒ J r ( D ) = J r ( D ′ ) . (cid:3) Rigid R-matrices from racks A binary relation from a set X to a set Y as a subset of the cartesian product X × Y . The composition of two binary relations R ⊂ X × Y and S ⊂ Y × Z is thebinary relation S ◦ R ⊂ X × Z defined by(57) S ◦ R := { ( x, z ) ∈ X × Z | ∃ y ∈ Y : ( x, y ) ∈ R, ( y, z ) ∈ S } . A span from a set X to a set Y is a triple U = ( U, s U , t U ) where U is a set ands U : U → X and t U : U → Y are set theoretical maps. The composition of twospans U from X to Y and V from Y to Z is the span from X to Z defined asthe pullback space (fibered product) V ◦ U := U × Y V together with the naturalprojections to X and Z .Two spans U and V from X to Y are called equivalent if there exists a bijection f : U → V such that s V ◦ f = s U and t V ◦ f = t U . The composition of spans inducesan associative binary operation for the equivalence classes of spans.Any binary relation R ⊂ X × Y is a special case of a span with s R : R → X andt R : R → Y being the canonical projections.Let Set be the monoidal category of sets with the cartesian product as themonoidal product, and
Rel (respectively
Span ) the extension of
Set with mor-phisms given by binary relations (respectively equivalence classes of spans). For amorphism Z : X → Y in Span , and any ( x, y ) ∈ X × Y , we denote(58) Z ( x, y ) := s − Z ( x ) ∩ t − Z ( y ) . We have a canonical monoidal functor(59) ̟ : Span → Rel which is identity on the level of objects and for any morphism Z : X → Y in Span ,the corresponding morphism in
Rel is given by(60) ̟ ( Z ) = { ( x, y ) ∈ X × Y | Z ( x, y ) = ∅} . Notice that if Z is a relation (as a particular case of spans) then ̟ ( Z ) = Z .Given a set theoretical map f : X → Y , its graph(61) Γ f := { ( x, f ( x )) | x ∈ X } ⊂ X × Y is naturally interpreted as a morphism ρ ( f ) in Rel and a morphism σ ( f ) in Span .The advantage of the categories
Rel and
Span over
Set is their rigidity, namely,for any set X , the diagonal ∆ X := Γ id X , interpreted as morphisms ε X : X × X → { } and η X : { } → X × X in Rel and
Span , gives rise to a canonical adjunction(
X, X, ε X , η X ) both in Rel and
Span .Let X be a (left) rack [4, 17, 7, 13], that is a set with a map X → X , ( x, y ) ( x · y, x ∗ y ) , NVARIANTS OF LONG KNOTS 9 such that the binary operation x · y is left self-distributive x · ( y · z ) = ( x · y ) · ( x · z ) , ∀ ( x, y, z ) ∈ X , and x ∗ ( x · y ) = y, ∀ ( x, y ) ∈ X . It is easily verified that for any rack X , the set-theoretical map r : X → X , ( x, y ) ( x · y, x ) , is a rigid R-matrix in the categories Rel and
Span . Moreover, all the relevantmorphisms are realised by set-theoretical maps. Indeed, introducing the maps r ′ , s, s ′ : X → X , r ′ ( x, y ) = ( y, y · x ) , s ( x, y ) = ( x ∗ y, x ) , s ′ ( x, y ) = ( y, y ∗ x ) , we obtain r − = e r = s ′ , ( e r ) − = r, g r − = ee r = s, (cid:16)g r − (cid:17) − = gg r − = r ′ . Thus, we obtain two long knot invariants J ρ ( r ) ( D ) in Rel and J σ ( r ) ( D ) in Span which are related to each other by the equality(62) J ρ ( r ) ( D ) = ̟ ( J σ ( r ) ( D )) . Racks associated to pointed groups.
Let (
G, µ ) be a pointed group thatis a group G together with a fixed element µ ∈ G . Then, it is easily verified thatthe set G with the map(63) G × G → G × G, ( g, h ) ( gµg − h, gµ − g − h )is a rack, and, thus, it gives rise to a rigid R-matrix(64) r G,µ : G × G → G × G, ( g, h ) ( gµg − h, g ) , both in the categories Rel and
Span . In this way, we obtain two long knot invari-ants J ρ ( r G,µ ) ( D ) and J σ ( r G,µ ) ( D ) related to each other by the equality(65) J ρ ( r G,µ ) ( D ) = ̟ ( J σ ( r G,µ ) ( D )) . Theorem 2.
There exists a canonical choice of a meridian-longitude pair ( m, ℓ ) oflong knots such that the set ( J σ ( r G,µ ) ( D ))(1 , λ ) is in bijection with the set of grouphomomorphisms (66) { h : π ( R \ f ( R ) , x ) → G | h ( m ) = µ, h ( ℓ ) = λ } where f : R → R is a long knot represented by D .Proof. Let f : R → R be a long knot whose image under the projection(67) p : R → R , ( x, y, z ) ( y, z ) . is the diagram ˜ D := ˙ D ◦ ξ − w ( ˙ D ) / with linearly ordered (from bottom to top) set ofarcs a , a , . . . , a n . As a result, the set of crossings acquires a linear order as well { c i | ≤ i ≤ n } where c i is the crossing separating the arcs a i − and a i and withthe over passing arc a κ i for a uniquely defined map(68) κ : { , . . . , n } → { , . . . , n } . Let t , t , . . . , t n ∈ R be a strictly increasing sequence such that f ( t ) = (0 , , t )for all t [ t , t n ], and for each i ∈ { , . . . , n − } , p ( f ( t i )) belongs to arc a i and is distinct from any crossing. Choose a base point x = ( s, ,
0) with sufficiently large s ∈ R > , a sufficiently small ǫ ∈ R > , and define the following paths(69) α , β i , γ i : [0 , → R , i ∈ { , . . . , n } , α ( t ) = ( ǫ cos(2 πt ) , − ǫ sin(2 πt ) , t ) ,β i ( t ) = (1 − t ) x + ( f ( t i ) + ( ǫ, , t, γ i ( t ) = f ((1 − t ) t i + tt ) + ( ǫ, , . To each arc a i of ˜ D , we associate the homotopy class(70) e i := [ β i · γ i · ¯ β ] ∈ π ( R \ f ( R ) , x ) , so that e = 1, and the Wirtinger generator(71) w i := [ β i · γ i · α · ¯ γ i · ¯ β i ] ∈ π ( R \ f ( R ) , x ) . We have the equalities(72) w i = e i w e − i , ∀ i ∈ { , , . . . , n } , (73) e i = w ε i κ i e i − , ∀ i ∈ { , . . . , n } , where ε i ∈ {± } is the sign of the crossing c i , and(74) e i = w ε i κ i w ε i − κ i − · · · w ε κ , ∀ i ∈ { , . . . , n } . We define the canonical meridian-longitude pair ( m, ℓ ) as follows(75) m := w , ℓ := e n . Taking into account the condition P ni =1 ε i = 0, we see that ℓ has the trivial imagein H ( R \ f ( R ) , Z ).Let us show that the following finitely presented groups are isomorphic to theknot group π ( R \ f ( R ) , x ):(76) E := h m, e , . . . , e n | e = 1 , e i = e κ i m ε i e − κ i e i − , ≤ i ≤ n i and(77) W := h w , . . . , w n | w ε i κ i w i − = w i w ε i κ i , ≤ i ≤ n i . As W is nothing else but the Wirtinger presentation of π ( R \ f ( R ) , x ), it sufficesto see the isomorphism E ≃ W . To see the latter, we remark that there are twogroup homomorphisms(78) u : W → E, w i e i me − i , i ∈ { , , . . . , n } , and(79) v : E → W, m w , e , e i w ε i κ i w ε i − κ i − · · · w ε κ , ∀ i ∈ { , . . . , n } . Indeed, we have(80) u ( w κ i ) ε i u ( w i − ) = u ( w i ) u ( w κ i ) ε i ⇔ u ( w κ i ) ε i e i − me − i − = e i me − i u ( w κ i ) ε i ⇔ e − i u ( w κ i ) ε i e i − m = me − i u ( w κ i ) ε i e i − ⇐ e − i u ( w κ i ) ε i e i − = 1 ⇔ e i = u ( w κ i ) ε i e i − ⇔ e i = e κ i m ε i e − κ i e i − implying that u is a group homomorphism. We also have(81) v ( e i ) = v ( e κ i ) v ( m ) ε i v ( e κ i ) − v ( e i − ) ⇔ v ( e i ) v ( e i − ) − = v ( e κ i ) w ε i v ( e κ i ) − ⇔ w κ i = v ( e κ i ) w v ( e κ i ) − ⇐ { w i = v ( e i ) w v ( e i ) − } ≤ i ≤ n NVARIANTS OF LONG KNOTS 11 and, for all i ∈ { , . . . , n } ,(82) v ( e i ) w = w ε i κ i · · · w ε κ w = w ε i κ i · · · w ε κ w w ε κ = . . . = w ε i κ i · · · w ε k κ k w k − w ε k − κ k − · · · w ε κ = · · · = w i w ε i κ i · · · w ε κ = w i v ( e i )implying that v is a group homomorphism as well.It remains to show that v ◦ u = id W and u ◦ v = id E . Indeed, for all i ∈ { , . . . , n } ,we have(83) v ( u ( w i )) = v ( e i me − i ) = v ( e i ) v ( m ) v ( e i ) − = v ( e i ) w v ( e i ) − = w i implying that v ◦ u = id W . We prove that u ( v ( e i )) = e i for all i ∈ { , . . . , n } byrecursion on i . For i = 0, we have(84) u ( v ( e )) = u (1) = 1 = e . Assuming that u ( v ( e k − )) = e k − for some k ∈ { , . . . , n − } , we calculate(85) u ( v ( e k )) = u ( w ε k κ k v ( e k − )) = u ( w κ k ) ε k u ( v ( e k − )) = e κ k m ε k e − κ k e k − = e k . Now, any element g of the set ( J σ ( r G,µ ) ( D ))(1 , λ ) is a map(86) g : { , , . . . , n } → G such that(87) g = 1 , g n = λ, g i = g κ i µ ε i g − κ i g i − , ∀ i ∈ { , . . . , n } . That means that g determines a unique group homomorphism(88) h g : E → G such that h g ( m ) = µ and h g ( e i ) = g i for all i ∈ { , , . . . , n } . On the other hand,any group homomorphism h : E → G such that h ( m ) = µ and h ( ℓ ) = λ is of theform h = h g where g i = h ( e i ). Thus, the map g h g is a set-theoretical bijectionbetween ( J σ ( r G,µ ) ( D ))(1 , λ ) and the set of group homomorphisms (66). (cid:3) Remark 4.
Theorem 2 illustrates the importance of considering long knots asopposed to closed knots. Namely, by closing a long knot, one identifies two openstrands and all the associated data. In particular, one has to impose the equality λ = 1 that corresponds to considering only those representations where the longi-tude is realized trivially. That means that in the case of closed diagrams one wouldobtain less powerful invariants.3.2. An extended Heisenberg group.
Let R be a commutative unital ring and G , the subgroup of GL (3 , R ) given by upper triangular matrices of the form(89) a b d c a , a ∈ U ( R ) , ( b, c, d ) ∈ R , where U ( R ) the group of units of R . Then, the elements(90) µ = t t , λ = s , commute with each other for any t ∈ U ( R ) and s ∈ R . Given the fact that G is analgebraic group, the invariant ( J ρ ( r G,µ ) ( D ))(1 , λ ) factorises through an ideal I D inthe polynomial algebra Q [ t, t − , s ]. Examples of calculations show that this ideal is often principal and is generated by the polynomial ∆ D s where ∆ D := ∆ D ( t ) is theAlexander polynomial of D . Nonetheless, there are examples for which this is notthe case, at least if the Alexander polynomial has multiple roots. In Table 1, wehave collected few selected examples of calculations for knots where the Alexanderpolynomials of the first three examples of knots 3 , 4 and 6 have simple roots,while for all other examples the respective Alexander polynomials have multipleroots and they are factorized into products of the Alexander polynomials of thefirst three examples. In particular, one can see that the knots 8 and 8 havedifferent Alexander polynomials but one and the same ideal, while the knots 8 and 9 have one and the same Alexander polynomial but different ideals.knot ∆ D I D − t + t (∆ s )4 − t + t (∆ s )6 − t + 3 t − t + t (∆ s )8 ∆ (∆ s, s )8 ∆ ∆ (∆ ∆ s )8 ∆ (∆ s, s )9 ∆ ∆ (∆ ∆ s, ∆ s )10 ∆ (∆ s, s )10 ∆ (∆ s )10 ∆ (∆ s, s )11 a ∆ (∆ s, s ) Table 1.
Examples of calculation with the extended Heisenberg group.4.
Invariants from Hopf algebras
Let
Hopf K be the category of Hopf algebras over a field K with invertible an-tipode. We have a contravariant endofunctor ( · ) o : Hopf K → Hopf K that asso-ciates to a Hopf algebra H with the multiplication ∇ its restricted dual (91) H o := ( ∇ ∗ ) − ( H ∗ ⊗ H ∗ )that can also be identified with the vector subspace of the algebraic dual H ∗ gen-erated by all matrix coefficients of all finite dimensional representations of H [3].Following [12], Drinfeld’s quantum double of H ∈ Ob Hopf K is a Hopf algebra D ( H ) ∈ Ob Hopf K uniquely determined by the property that there are two Hopfalgebra inclusions(92) ı : H → D ( H ) , : H o, op → D ( H )such that D ( H ) is generated by their images subject to the commutation relations(93) ( 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 comultiplication(94) ∆( 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 quasitriangular Hopf algebra with the dual universal R -matrix (95) ̺ : ( D ( H )) o ⊗ ( D ( H )) o → K , x ⊗ y
7→ h x, ( ı o ( y )) i NVARIANTS OF LONG KNOTS 13 which, among other things, satisfies the Yang–Baxter relation(96) ̺ , ∗ ̺ , ∗ ̺ , = ̺ , ∗ ̺ , ∗ ̺ , in the convolution algebra ((( D ( H )) o ) ⊗ ) ∗ , and any finite-dimensional right comod-ule(97) V → V ⊗ ( D ( H )) o , v v (0) ⊗ v (1) , gives rise to a rigid R -matrix(98) r V : V ⊗ V → V ⊗ V, u ⊗ v v (0) ⊗ u (0) h ̺, u (1) ⊗ v (1) i . This implies that there exists a universal invariant of long knots Z H ( K ) taking itsvalues in the convolution algebra (( D ( H )) o ) ∗ such that(99) J r V ( K ) v = v (0) h Z H ( K ) , v (1) i , ∀ v ∈ V. Remark 5.
By using the coend of the monoidal braided category of finite dimen-sional comodules over ( D ( H )) o , the universal invariant Z H ( K ) can be interpretedvia the Lyubashenko theory [11]. In the case of finite dimensional quasitriangularHopf algebras, this coend approach is further developed by Virelizier in [18]. Remark 6. If H is a finite-dimensional Hopf algebra with a linear basis { e i } i ∈ I and { e i } i ∈ I is the dual linear basis of H ∗ , then, the dual universal R -matrix isconjugate to the universal R -matrix(100) R := X i ∈ I ( e i ) ⊗ ı ( e i ) ∈ D ( H ) ⊗ D ( H )in the sense that, for any x, y ∈ ( D ( H )) o = ( D ( H )) ∗ , we have(101) h x ⊗ y, R i = X i ∈ I h x, ( e i ) ih y, ı ( e i ) i = X i ∈ I h x, ( e i ) ih ı o ( y ) , e i i = * x, X i ∈ I h ı o ( y ) , e i i e i !+ = h x, ( ı o ( y )) i = h ̺, x ⊗ y i . In the infinite-dimensional case, formula (100) is formal but it is a convenientand useful tool for actual calculations.
Remark 7.
The algebra inclusion D ( H ) ⊂ (( D ( H )) o ) ∗ allows to think of theconvolution algebra (( D ( H )) o ) ∗ as a certain algebra completion of the quantumdouble D ( H ).4.1. A Hopf algebra associated to a two-dimensional Lie group.
Let B bethe commutative Hopf algebra over C generated by an invertible group-like element a and element b with the coproduct(102) ∆( b ) = a ⊗ b + b ⊗ { b m a n | ( m, n ) ∈ Z ≥ × Z } is a linear basis of B .The restricted dual Hopf algebra B o is generated by two primitive elements ψ and φ and group-like elements(103) u ψ , e vφ , ( u, v ) ∈ C =0 × C , which satisfy the commutation relations(104) [ ψ, φ ] = φ, [ ψ, e vφ ] = vφe vφ , u ψ φu − ψ = uφ, u ψ e vφ u − ψ = e uvφ , [ ψ, u ψ ] = [ φ, e vφ ] = 0 , u ψ z ψ = ( uz ) ψ , e vφ e wφ = e ( v + w ) φ . As linear forms on B , they are defined by the relations(105) h ψ, b m a n i = δ m, n, h φ, b m a n i = δ m, , h u ψ , b m a n i = δ m, u n , h e vφ , b m a n i = v m , ∀ ( m, n ) ∈ Z ≥ × Z . Using the notation ˙ x := ı ( x ) for x ∈ { a, b } , ˙ y := ( y ) for y ∈ { ψ, φ } and(106) u ˙ ψ := ( u ψ ) , e v ˙ φ := ( e vψ ) , the commutation relations (93) in the case of the quantum double D ( B ) take theform(107) [ ˙ ψ, ˙ b ] = ˙ b, [ ˙ φ, ˙ b ] = 1 − ˙ a, u ˙ ψ ˙ bu − ˙ ψ = u ˙ b, e v ˙ φ ˙ be − v ˙ φ = ˙ b + (1 − ˙ a ) v and ˙ a is central. The formal universal R -matrix that reads as(108) R := (1 ⊗ ˙ a ) ˙ ψ ⊗ e ˙ φ ⊗ ˙ b = X m,n ≥ n ! (cid:18) ˙ ψm (cid:19) ˙ φ n ⊗ ( ˙ a − m ˙ b n should be interpreted as follows.Any finite dimensional right comodule V over ( D ( B )) o is a left module over D ( B ) defined by(109) xv = v (0) h v (1) , x i , ∀ ( x, v ) ∈ D ( B ) × V. Thus, it suffices to make sense of formula (108) in the case of an arbirary finite-dimensional representation of D ( B ) where the elements 1 − ˙ a , ˙ b and ˙ φ are necessarilynilpotent, so that the formal infinite double sum truncates to a well defined finitesum. Conjecture 1.
The universal invariant associated to the Hopf algebra B is ofthe form Z B ( K ) = (∆ K ( ˙ a )) − where ∆ K ( t ) is the Alexander polynomial of K normalised so that ∆ K (1) = 1 and ∆ K ( t ) = ∆ K (1 /t ).By direct computation, we were able to prove this conjecture in the case of thetrefoil knot. Justification in the general case comes from the following reasoning.The Hopf algebra B can be q -deformed to a non-commutative Hopf algebra B q with the same coalgebra structure (102) but with q -comutative relation ab = qba .We have B = B . For q not a root of unity, the quantum double D ( B q ) is closelyrelated to the quantum group U q ( sl ). In particular, for each n ∈ Z > , it admits an n -dimensional irreducible representation corresponding to the n -th colored Jonespolynomial. In the limit n → ∞ with q = t /n and fixed t , one recovers aninfinite-dimensional representation of the Hopf algebra B where the central element˙ a takes the value t . On the other hand, according to the Melvin–Morton–Rozanskyconjecture proven by Bar-Nathan and Garoufalidis in [1] and by Garoufalidis andLˆe in [5], the n -th colored Jones polynomial in that limit tends to (∆ K ( t )) − . NVARIANTS OF LONG KNOTS 15
References [1] Dror Bar-Natan and Stavros Garoufalidis. On the Melvin-Morton-Rozansky conjecture.
In-vent. Math. , 125(1):103–133, 1996.[2] Alain Brugui`eres and Alexis Virelizier. Hopf diagrams and quantum invariants.
Algebr. Geom.Topol. , 5:1677–1710 (electronic), 2005.[3] Sorin D˘asc˘alescu, Constantin N˘ast˘asescu, and S¸erban Raianu.
Hopf algebras , volume 235of
Monographs and Textbooks in Pure and Applied Mathematics . Marcel Dekker, Inc., NewYork, 2001. An introduction.[4] Roger Fenn and Colin Rourke. Racks and links in codimension two.
J. Knot Theory Ramifi-cations , 1(4):343–406, 1992.[5] Stavros Garoufalidis and Thang T. Q. Lˆe. Asymptotics of the colored Jones function of aknot.
Geom. Topol. , 15(4):2135–2180, 2011.[6] Kazuo Habiro. Bottom tangles and universal invariants.
Algebr. Geom. Topol. , 6:1113–1214,2006.[7] David Joyce. A classifying invariant of knots, the knot quandle.
J. Pure Appl. Algebra ,23(1):37–65, 1982.[8] R. M. Kashaev. R -matrix knot invariants and triangulations. In Interactions between hyper-bolic geometry, quantum topology and number theory , volume 541 of
Contemp. Math. , pages69–81. Amer. Math. Soc., Providence, RI, 2011.[9] R. J. Lawrence. A universal link invariant using quantum groups. In
Differential geometricmethods in theoretical physics (Chester, 1988) , pages 55–63. World Sci. Publ., Teaneck, NJ,1989.[10] H. C. Lee. Tangles, links and twisted quantum groups. In
Physics, geometry, and topology(Banff, AB, 1989) , volume 238 of
NATO Adv. Sci. Inst. Ser. B Phys. , pages 623–655. Plenum,New York, 1990.[11] Volodimir Lyubashenko. Tangles and Hopf algebras in braided categories.
J. Pure Appl.Algebra , 98(3):245–278, 1995.[12] Shahn Majid.
Foundations of quantum group theory . Cambridge University Press, Cambridge,1995.[13] S. V. Matveev. Distributive groupoids in knot theory.
Mat. Sb. (N.S.) , 119(161)(1):78–88,160, 1982.[14] Tomotada Ohtsuki. Colored ribbon Hopf algebras and universal invariants of framed links.
J. Knot Theory Ramifications , 2(2):211–232, 1993.[15] N. Yu. Reshetikhin. Quasitriangular Hopf algebras and invariants of links.
Algebra i Analiz ,1(2):169–188, 1989.[16] N. Yu. Reshetikhin and V. G. Turaev. Ribbon graphs and their invariants derived fromquantum groups.
Comm. Math. Phys. , 127(1):1–26, 1990.[17] Mituhisa Takasaki. Abstraction of symmetric transformations.
Tˆohoku Math. J. , 49:145–207,1943.[18] Alexis Virelizier. Kirby elements and quantum invariants.
Proc. London Math. Soc. (3) ,93(2):474–514, 2006.
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