Quantum traces for \mathrm{SL}_n(\mathbb{C}): the case n=3
QQUANTUM TRACES FOR SL n ( C ) : THE CASE n = 3 DANIEL C. DOUGLAS
Abstract.
We generalize Bonahon and Wong’s SL ( C )-quantum trace map to the settingof SL ( C ). More precisely, for each non-zero complex number q , we associate to everyisotopy class of framed oriented links K in a thickened punctured surface S × (0 ,
1) a Laurentpolynomial Tr qλ ( K ) = Tr qλ ( K )( X qi ) in q -deformations X qi of the Fock-Goncharov coordinates X i for a higher Teichm¨uller space, depending on the choice of an ideal triangulation λ of thesurface S . Along the way, we propose a definition for a SL n ( C )-version of this invariant. For a finitely generated group Γ and a suitable Lie group G , a primary object of study inlow-dimensional geometry and topology is the G -character variety X G (Γ) = { ρ : Γ −! G } // G consisting of group homomorphisms ρ from Γ to G , considered up to conjugation. Here, thequotient is taken in the algebraic geometric sense of Geometric Invariant Theory [MFK94].Character varieties can be explored using a wide variety of mathematical skill sets. Someexamples include the Higgs bundle approach of Hitchin [Hit92], the dynamics approach ofLabourie [Lab06], and the representation theory approach of Fock and Goncharov [FG06b].In the case where the group Γ = π ( S ) is the fundamental group of a punctured surface S of finite topological type, and where the Lie group G = SL n ( C ) is the special linear group,we are interested in building a bridge between two competing deformation quantizations ofthe character variety X SL n ( C ) ( S ) := X SL n ( C ) ( π ( S )). Here, a deformation quantization { X q } q of a Poisson space X is a family of non-commutative algebras X q parametrized by a non-zerocomplex parameter q = e πi (cid:126) , such that the lack of commutativity in X q is infinitesimallymeasured in the semi-classical limit (cid:126) ! X . In thecase where X is the character variety X SL n ( C ) ( S ), the Poisson bracket is provided by theGoldman-Weil-Petersson Poisson structure on the character variety [Gol84, Gol86].The first quantization of the character variety is the SL n ( C )-skein algebra S qn ( S ) of the sur-face S , developed by several groups of researchers: by [Tur88, Tur89, Wit89, Prz91, Tur91,Bul97a, Bul97b, BFKB99, PS00, CM12], based on [Con70, Kau87], in the case of SL ( C ); by[Jae92, Kup96, OY97] in the case of SL ( C ); and, by [Tur89, MOY98, Sik01, Sik05, CKM14]in the case of SL n ( C ). The skein algebra is motivated by the classical algebraic geomet-ric approach to studying the character variety X SL n ( C ) ( S ) via its commutative algebra ofregular functions C [ X SL n ( C ) ( S )]. An example of a regular function is the trace functionTr γ : X SL n ( C ) ( S ) ! C associated to an oriented closed curve γ ∈ π ( S ) sending a repre-sentation ρ : π ( S ) ! SL n ( C ) to the trace Tr( ρ ( γ )) ∈ C of the matrix ρ ( γ ) ∈ SL n ( C ).A theorem of Classical Invariant Theory, by Procesi [Pro76], says that the trace functionsTr γ generate the algebra of functions C [ X SL n ( C ) ( S )] as an algebra, and it also identifies all Date : January 19, 2021.This work was partially supported by the U.S. National Science Foundation grants DMS-1107452,1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network), andalso by the U.S. National Science Foundation grants DMS-1406559 and DMS-1711297. a r X i v : . [ m a t h . G T ] J a n DANIEL C. DOUGLAS of the relations. According to the philosophy of Turaev and Witten, quantizations of thecharacter variety, such as the skein algebra S qn ( S ), should be of a 3-dimensional nature.Indeed, elements of S qn ( S ) are represented by linear combinations of framed oriented links K in the 3-dimensional thickened surface S × (0 , q and the integer n , such as theHOMFLYPT relation from knot theory [FYH +
85, PT87]. The skein algebra S qn ( S ) has theadvantage of being natural, but is difficult to work with in practice.The second quantization, due to Chekhov and Fock [Foc97, FC99] and Kashaev [Kas98]in the case of SL ( C ), and due to Fock and Goncharov [FG06a, FG06b, FG09] in the caseof SL n ( C ), is the quantum SL n ( C )-character variety (cid:98) T qn ( S ). At the classical level, Fock andGoncharov [FG06b] introduced a framed version X PSL n ( C ) ( S ) FG of the PSL n ( C )-character va-riety, whose points can roughly be thought of as points of the character variety X PSL n ( C ) ( S )equipped with additional linear algebraic data attached to the punctures of S . Associated toeach ideal triangulation λ of the punctured surface S is a coordinate chart U λ ∼ = ( C − { } ) N for X PSL n ( C ) ( S ) FG parametrized by N non-zero complex coordinates X , X , . . . , X N wherethe integer N depends only on the topology of the surface S and the rank of the Lie groupSL n ( C ). By lifting from PSL n ( C ) to SL n ( C ), a choice of coordinates X i determines anequivalence class of a representation ρ ( X i ) : π ( S ) ! SL n ( C ) well-defined up to the actionof the cohomology group H ( S , Z /n Z ). Thus, for each oriented closed curve γ ∈ π ( S ),the trace Tr( ρ ( X i )( γ )) ∈ C is well-defined up to multiplication by a n -root of unity. Theyalso constructed, for each immersed oriented closed curve γ (cid:48) in S , a Laurent polynomial (cid:102) Tr( γ (cid:48) )( X ± /ni ) ∈ C [ X ± /n , . . . , X ± /nN ] in formal n -roots of the variables X i , which we callthe classical trace polynomial associated to the immersed curve γ (cid:48) . This satisfies the propertythat if one chooses n -roots X /ni ∈ C −{ } of given coordinates X i , and if γ ∈ π ( S ), then theabove trace Tr( ρ ( X i )( γ )) ∈ C is equal to the evaluated Laurent polynomial (cid:102) Tr( γ (cid:48) )( X ± /ni )up to a n -root of unity, for any immersed representative γ (cid:48) of γ . At the quantum level,Fock and Goncharov defined, for each ideal triangulation λ of S , a non-commutative algebra (cid:98) T qn ( λ ) consisting of rational fractions in formal n -roots of q -deformed versions X qi of theircoordinates X i , which no longer commute but q -commute. The quantum character variety (cid:98) T qn ( S ) associated to S is obtained by gluing together these algebras (cid:98) T qn ( λ ) by rational tran-sition maps. In practice, we work with the quantum torus sub-algebra T qn ( λ ) ⊆ (cid:98) T qn ( λ ) ofLaurent polynomials in (cid:98) T qn ( λ ). The quantum character variety (cid:98) T qn ( S ) has the advantage ofbeing easier to work with than the skein algebra S qn ( S ), however it is less intrinsic.It is natural to seek a q -deformed version Tr qγ of the trace functions, assigning to a closedcurve γ a Laurent polynomial in the q -deformed Fock-Goncharov coordinates X qi . Indeed,the existence of quantum holonomies is implicit in [Wit89]. Following Turaev and Witten’sphilosophy, to construct a quantum trace one is led from the 2-dimensional setting of curves γ on the surface S to the 3-dimensional setting of knots K in the thickened surface S × (0 , Conjecture 1 (Quantum trace map) . Fix a n -root ω = q /n ∈ C − { } of q . For each idealtriangulation λ of the punctured surface S , there exists an injective algebra homomorphism Tr ωλ : S qn ( S ) (cid:44) −! T ωn ( λ ) from the skein algebra S qn ( S ) of the surface S to the Fock-Goncharov quantum torus T ωn ( λ ) associated to the ideal triangulation λ , satisfying the property that if q = ω = 1 and if K is UANTUM TRACES 3 a blackboard-framed oriented link with (cid:96) components in the thickened surface S × (0 , , then Tr λ ([ K ]) = (cid:96) (cid:89) j =1 (cid:102) Tr( γ j )( X ± /ni ) ∈ T n ( λ ) = C [ X ± /n , X ± /n , . . . , X ± /nN ] recovers from the skein [ K ] the product of the classical trace polynomials (cid:102) Tr( γ j )( X ± /ni ) associated to the immersed oriented closed curves γ j in S obtained by projecting to S ∼ = S × { / } the components K j in S × (0 , of the blackboard-framed oriented link K .Moreover, this assignment is natural with respect to the action of the mapping class groupof the surface S . More precisely, if λ (cid:48) is a different choice of ideal triangulation of S and if Θ ωλλ (cid:48) : (cid:98) T ωn ( λ (cid:48) ) −! (cid:98) T ωn ( λ ) is the corresponding rational transition map between the fraction algebras (cid:98) T ωn ( λ (cid:48) ) and (cid:98) T ωn ( λ ) ,then for each skein [ K ] ∈ S qn ( S ) the rational map Θ ωλλ (cid:48) sends the Laurent polynomial Tr ωλ (cid:48) ([ K ]) to the Laurent polynomial Tr ωλ ([ K ]) . Conjecture 1 is due to Chekhov and Fock [Foc97, CF00] in the case n = 2, and wasproved by hand in that case by Bonahon and Wong [BW11b]. For other work constructingquantum traces, making connections to Gaiotto, Moore, and Neitzke’s theory of spectralnetworks [GMN13], see [Gab17, KLS20, NY20].In Theorem 20, we prove a slightly weaker version of Conjecture 1 in the case n = 3. Theorem 2.
Fix a -root ω = q / ∈ C − { } of q . For each ideal triangulation λ of thepunctured surface S , there exists a mapping Tr ωλ : { isotopy classes of framed oriented links K in S × (0 , } −! T ω ( λ ) satisfying the property that if q = ω = 1 and if K is a blackboard-framed oriented link, then Tr λ ( K ) = (cid:96) (cid:89) j =1 (cid:102) Tr( γ j )( X ± / i ) ∈ T ( λ ) = C [ X ± / , X ± / , . . . , X ± / N ] . In addition, this assignment satisfies the
HOMFLYPT skein relation for SL ( C ) . Our proof is also by hand, generalizing the strategy of Bonahon and Wong. In particular,we rely on computer assistance to do some of the calculations, whose additional complexityowes to our working in the higher rank setting; see Appendix A. Along the way, we alsopropose a definition for the mapping Tr ωλ in the more general setting of SL n ( C ). In thatsetting, the obvious generalization of Theorem 2, namely the statement obtained by replacing3 with n everywhere in Theorem 2, should also hold. The difficulty in proving this moregeneral version is more a matter of complexity than of definitions. Its proof depends onestablishing a host of algebraic identities about n × n matrices with coefficients in the non-commutative algebra T ωn ( λ ). This is where a computer comes in handy in the case n = 3.Bonahon and Wong’s construction in the case n = 2 is implicitly related to the theory ofthe quantum group U q ( sl n ) or, more precisely, of its Hopf dual SL qn , which is also known asthe quantized coordinate ring O q (SL n ); see [Mon93, Kas95, Maj95, KS97, BG02]. However,they do not make this connection explicit. In some sense, this was possible due to the highlysymmetric nature of SL ( C ). On the other hand, in order to construct the SL n ( C )-quantumtrace map Tr ωλ one is compelled to more fully embrace the theoretical aspects of quantumgroups. Moreover, it appears that there are deeper interactions at play between quantum DANIEL C. DOUGLAS group theory and higher Teichm¨uller theory, which make the construction possible. In acompanion paper [Dou] we explore one such interaction, by establishing a local buildingblock result that explains conceptually, in terms of the dual quantum group SL qn , some ofthe aforementioned non-commutative identities in the Fock-Goncharov quantum torus T ωn ( λ );see Theorem 14. For another recent and independent appearance of these identities in thecontext of quantum integrable systems, see [CS20], motivated by [SS19, SS17]. In particular,their point of view should lead to a proof of the general SL n ( C )-version of Theorem 2. Forother related works in Poisson and quantum geometry, see [FG06a, FG09, GSV09, GS19].It remains to upgrade Theorem 2, and its SL n ( C )-generalization, to Conjecture 1. Thisposes a number of challenges, some of which we discuss now.We recall that an element of the skein algebra S qn ( S ) is represented by a formal linearcombination of framed oriented links K in S × (0 , ωλ fromTheorem 2 can be extended by linearity to define a mapping, also called Tr ωλ , from the skeinalgebra S qn ( S ) to the Fock-Goncharov quantum torus T ωn ( λ ). In order for this extended mapTr ωλ to be well-defined, one has to check that all of the skein relations in S qn ( S ) are satisfied.In Theorem 2, we check this only for the HOMFLYPT relation. It turns out that, for n > K ,but webs W , which are certain n -valent graphs embedded in S × (0 , W represents an element [ W ] of the skein algebra S qn ( S ).Note that, in particular, the skein [ W ] can be represented as a linear combination of links K .Thus, to check that the mapping Tr ωλ satisfies the skein relations, it is desirable to be able tocompute Tr ωλ ([ W ]) directly from the web W without re-writing W in terms of links. In fact,this computation of the quantum trace Tr ωλ ([ W ]) for webs W comes automatically from theconstruction of the Reshetikhin-Turaev invariant [RT90] for the dual quantum group SL qn .Independently, Lˆe [Lˆe19, Lˆe18, CL19] also recently [Lˆe] constructed a SL n ( C )-quantumtrace map Tr ωλ : S qn ( S ) ! T ωn ( λ ). His approach applies the upcoming joint work [LS] withSikora on SL n ( C )-stated skein algebras; see also [Hig20]. A slightly more general constructionthan ours, Lˆe defines his quantum trace map Tr ωλ ([ W ]) for webs W in S × (0 , K , his map agrees (possibly up to choices of conventions) with themap for SL n ( C ) that we define in this article. Since, as discussed previously, webs can bewritten as a linear combination of links in the skein algebra S qn ( S ), it follows that our twomaps Tr ωλ agree on the whole skein algebra. Also recently, in the case n = 3, Kim [Kim20]extended our map Tr ωλ from links to webs; for another related upcoming work, see [IY].Another challenge is to establish the injectivity of the mapping Tr ωλ in Conjecture 1. In fact,this is essentially a problem about the classical geometry of the character variety. Indeed, thisproperty is not even evident for q = 1, and in the case of SL ( C ) the proof for arbitrary q ismore or less the same as the proof for q = 1; see [BW11b, § n ( C ),by employing the same strategy as for SL ( C ) one quickly arrives at the well-known problemof explicitly constructing and coordinatizing a linear basis, indexed by planar web diagramson S , for the algebra of functions C [ X SL n ( C ) ( S )] on the character variety when q = 1, oressentially equivalently for the skein algebra S qn ( S ) when q is arbitrary; see [Prz91, HP93,Thu14] for the n = 2 case, and [Kup96, SW07, Fon12, FKK13, CKM14] for the higher rankcase. This problem is also intimately related to the so-called classical and quantum Fock-Goncharov Duality Conjecture [FG06b, FG09, GS15, GHKK18, GS18, GS19]. For recentprogress on these closely related problems, see [FS20, Dou20, DS20a, Kim20, DS20b, Sun].In particular, the work [Dou20, DS20a, DS20b, Sun], motivated by [Kup96, Xie13, GS15], UANTUM TRACES 5 develops the key ingredient going into the proof of the injectivity of the mapping Tr ωλ , namelyconstructing coordinates for webs, and is also related to the naturality property appearing inConjecture 1; as a future application, these works, together with the SL n ( C )-quantum tracemap Tr ωλ , should yield a construction of a linear basis for the SL n ( C )-skein algebra S qn ( S ).Bonahon and Wong [BW16, BW11a] applied the SL ( C )-quantum trace map as a toolto explore the representation theory of the Kauffman bracket skein algebra S q ( S ), which isrelevant to the study of quantum invariants of 3-dimensional manifolds [Wit89, RT91, Lic93,BHMV95]; more recently, see [FKBL19, GJS19a, GJS19b]. In addition, the quantum tracemap can be used to give simple proofs of fundamental properties of skein algebras that areoften difficult to prove otherwise; see [AF17, PS19]. It was also an essential ingredient in[AK17] which studied the quantum Fock-Goncharov Duality Conjecture in the case n = 2,along the way demonstrating impressive positivity properties; see also [All19, CKKO20]. Weexpect that SL n ( C )-quantum traces can be similarly applied in the higher rank setting. Acknowledgements
This work would not have been possible without the constant support of my Ph.D. advisorFrancis Bonahon, to whom I am extremely grateful for the very many hours of his time thathe most likely could have better spent on “applied differential geometry” (a.k.a. hiking)rather than putting up with me. Many thanks also go out to Sasha Shapiro and Thang Lˆe,for informing me about related research and for enjoyable conversations, as well as to DylanAllegretti and Zhe Sun, who helped me fine-tune my ideas during a very pleasant visit toTsinghua University in Beijing, supported by a GEAR graduate internship grant.1.
Classical trace polynomial
We associate a Laurent polynomial (cid:102)
Tr( γ )( X ± /ni ) in commuting formal n -roots X ± /n , X ± /n , . . . , X ± /nN to each immersed oriented closed curve γ in the surface S , where N depends only on the topology of S and the rank of the Lie group SL n ( C ).1.1. Topological setup.
Let S be an oriented punctured surface of finite topological type,namely S is obtained by removing a finite subset P , called the set of punctures , from acompact oriented surface S . In particular, note that S may have non-empty boundary, ∂ S (cid:54) = ∅ . We require that there is at least one puncture, that each component of ∂ S is punctured, and that the Euler characteristic χ ( S ) of the punctured surface S satisfies χ ( S ) < d/ d is the number of components of ∂ S . Note that each component of ∂ S (a) Four times punctured sphere (b)
Once punctured torus
Figure 1.
Ideal triangulations ( ∂ S = ∅ ) DANIEL C. DOUGLAS is an ideal arc . These topological conditions guarantee the existence of an ideal triangulation λ of the punctured surface S , namely a triangulation λ of the surface S whose vertex set isequal to the set of punctures P . See Figure 1 for some examples of ideal triangulations. Theideal triangulation λ consists of (cid:15) = − χ ( S ) + 2 d edges E and τ = − χ ( S ) + d triangles T .To simplify the exposition, we always assume that λ does not contain any self-folded trian-gles . Consequently, each triangle T of λ has three distinct edges. Such an ideal triangulation λ always exists, except in a handful of small examples. Our results should generalize to allowfor self-folded triangles, requiring only minor adjustments.1.2. Discrete triangle.
The discrete n -triangle Θ n isΘ n = (cid:8) ( a, b, c ) ∈ Z ; a, b, c (cid:62) , a + b + c = n (cid:9) as shown in Figure 2.The interior of the n -discrete triangle isint(Θ n ) = (cid:8) ( a, b, c ) ∈ Z ; a, b, c > , a + b + c = n (cid:9) . Figure 2.
Discrete triangle ( n = 5)1.3. Dotted ideal triangulations.
Let the punctured surface S be equipped with an idealtriangulation λ , and let N = (cid:15) ( n −
1) + τ ( n − n − /
2; see § dotted ideal triangulation is the pair consisting of λ together with N distinct dots attached to the edges E and triangles T of λ , where there are n − edge-dots attachedto each edge E and ( n − n − / triangle-dots attached to each triangle T . For eachtriangle T together with its three boundary edges E , E , E , these dots are arranged as thevertices of the discrete n -triangle Θ n (minus its three corner vertices) overlaid on top of theideal triangle T ; see Figures 3 and 4. Note that we draw boundary-dots as small squares and interior-dots as small disks.Given a triangle T of λ , which inherits an orientation from the orientation of S , and givenan edge E of T , it makes sense to talk about whether one edge-dot on E is to the left or tothe right of another edge-dot on E as viewed from the triangle T ; see Figure 3b.We always assume that we have chosen an ordering for the N dots lying on the dottedideal triangulation λ , so that it makes sense to talk about the i -th dot, i = 1 , , . . . , N . UANTUM TRACES 7 (a)
Four times punctured sphere (b)
Ideal triangle
Figure 3.
Dotted ideal triangulations ( n = 3) Figure 4.
Dotted ideal triangulation ( n = 4)1.4. Classical polynomial algebra.
Let the punctured surface S be equipped with adotted ideal triangulation λ . Definition 3.
The classical polynomial algebra T n ( λ ) = C [ X ± /n , X ± /n , . . . , X ± /nN ] associ-ated to the dotted ideal triangulation λ is the commutative algebra of Laurent polynomialsgenerated by formal n -roots X /ni and their inverses. We think of the generator X /ni asassociated to the i -th dot lying on λ . As for dots, see § edge- and triangle-generators as well as boundary- and interior-generators . Elements X ± i = ( X ± /ni ) n of T n ( λ )are called coordinates . We often indicate edge-coordinates with the letter Z instead of X . Remark 4.
The coordinates X i in the classical polynomial algebra are, intuitively speaking,algebraizations of Fock and Goncharov’s geometric coordinates for the framed PSL n ( C )-character variety X PSL n ( C ) ( S ) FG ; see the Introduction. There is a caveat, which is that inthe classical geometric setting the Fock-Goncharov coordinates X i are only associated to theinterior-dots (in other words, not to the boundary-dots), whereas in the quantum algebraicsetting there are coordinates X i associated to the boundary-dots as well.In the theory of cluster algebras [FZ02], these boundary-variables are often called frozenvariables [FG06a, FG06b, FWZ16]. From the classical geometric point of view, one mightthink of the frozen boundary-coordinates as having the “potential” to become “actual” un-frozen interior-coordinates if the surface-with-boundary S were included inside the interiorof a larger surface S (cid:48) . It is perhaps surprising that, at the quantum algebraic level, theinclusion of boundary-coordinates is an essential conceptual step that needs to be taken inorder to witness the emergence of local quantum properties, as we will see later on. DANIEL C. DOUGLAS
Shearing matrices for edges and triangles.
Let M n ( T n ( λ )) denote the usual non-commutative algebra of n × n matrices with coefficients in the commutative classical polyno-mial algebra T n ( λ ) = C [ X ± /n , X ± /n , . . . , X ± /nN ]. Define the special linear group SL n ( T n ( λ ))to be the subset of M n ( T n ( λ )) consisting of the matrices with determinant equal to 1.Let Z = X ± i be an edge-coordinate in the classical polynomial algebra T n ( λ ). For j =1 , , . . . , n − j -th edge-shearing matrix S edge j ( Z ) ∈ SL n ( T n ( λ )) by S edge j ( Z ) = Z − j/n Z Z ... Z ... ∈ SL n ( T n ( λ )) ( Z appears j times) . Notice the normalizing factor Z − j/n multiplying the matrix on the left (or on the right).Similarly, for any triangle-coordinate X = X ± i in T n ( λ ) and for any index j = 1 , , . . . , n − j -th left triangle-shearing matrix S left j ( X ) ∈ SL n ( T n ( λ )) by S left j ( X ) = X − ( j − /n X ... X ... ∈ SL n ( T n ( λ )) ( X appears j − j -th right triangle-shearing matrix S right j ( X ) ∈ SL n ( T n ( λ )) by S right j ( X ) = X ( j − /n ... X − ... X − ∈ SL n ( T n ( λ )) ( X appears j − . Note that S left1 ( X ) and S right1 ( X ) do not, in fact, involve the variable X , and so we will denotethese matrices simply by S left1 and S right1 , respectively. Remark 5.
In the theory of Fock and Goncharov, these shearing matrices for edges andtriangles are called snake matrices . Each one is the coordinate transformation matrix passingbetween a pair of compatibly normalized projective coordinate systems associated to a pair ofadjacent snakes . For a framed local system in X PSL n ( C ) ( S ) FG with coordinates X i , computingthe monodromy of the local system around a curve γ amounts to multiplying together asequence of snake matrices along the direction of the curve. For more details, see [FG06b, § ( R ) or SL ( C ) theory, where the Fock-Goncharovcoordinates X i coincide with Thurston’s [Thu97] shearing or shear-bend coordinates forTeichm¨uller space; see [Foc97, FG07a, HN16] for more details. There is also a geometricinterpretation of Fock and Goncharov’s coordinates in the case n = 3, where they parametrizeconvex projective structures on the surface S ; see [FG07b, CTT20]. UANTUM TRACES 9
Local monodromy matrices.
Consider a dotted ideal triangle T , which we think ofas sitting inside a larger dotted ideal triangulation λ ; see Figure 3. We assign n × n matriceswith coefficients in the classical polynomial algebra T n ( λ ) = C [ X ± /n , X ± /n , . . . , X ± /nN ] tovarious “short” oriented arcs lying on the surface S .Recall that we think of the n -discrete triangle Θ n ( § T , so that there is a one-to-one correspondence between coordinates X i = X abc in T n ( λ ) andvertices ( a, b, c ) in Θ n − { ( n, , , (0 , n, , (0 , , n ) } (in other words, in the n -discrete triangleΘ n minus its three corner vertices). Note that X abc is a triangle-coordinate if and only if( a, b, c ) is an interior point ( a, b, c ) ∈ int(Θ n ), otherwise X abc is an edge-coordinate.We will use the following notation for writing products of matrices. Given a family M i of n × n matrices, put N (cid:89) i = M M i = M M M M +1 · · · M N , M (cid:89) i = N +1 M i = 1 ( M (cid:54) N ) , M (cid:97) i = N M i = M N M N − · · · M M , N (cid:97) i = M − M i = 1 ( M (cid:54) N ) . First, consider a left-moving arc γ , as shown in Figure 5. We assume γ has no kinks(Figures 18 and 19). Define the associated left matrix M left ( X i ’s) in SL n ( T n ( λ )) by M left ( X i ’s) = (cid:97) i = n − (cid:32) S left1 i (cid:89) j =2 S left j (cid:0) X ( j − n − i )( i − j +1) (cid:1)(cid:33) ∈ SL n ( T n ( λ ))where the matrix S left j ( X abc ) is the j -th left triangle-shearing matrix; see § Figure 5.
Left matrixNext, consider a right-moving arc γ , as shown in Figure 6. We assume γ has no kinks.We define the associated right matrix M right ( X i ’s) in SL n ( T n ( λ )) by M right ( X i ’s) = (cid:97) i = n − (cid:32) S right1 i (cid:89) j =2 S right j (cid:0) X ( i − j +1)( n − i )( j − (cid:1)(cid:33) ∈ SL n ( T n ( λ ))where the matrix S right j ( X abc ) is the j -th right triangle-shearing matrix; see § edge-crossing arc γ , as shown on the left hand or right hand sideof Figure 7. Let Z j , j = 1 , , . . . , n −
1, be the j -th edge-coordinate, measured from right to Figure 6.
Right matrixleft as seen by the triangle out of which the arc is moving. Define the associated edge matrix E ( Z j ’s) in SL n ( T n ( λ )) by E ( Z j ’s) = n − (cid:89) j =1 S edge j ( Z j ) ∈ SL n ( T n ( λ ))where the matrix S edge j ( Z j ) is the j -th edge-shearing matrix; see § Z j according to the rule Z j ↔ Z n − j ; see Figure 7.Observe that the order in which the shearing matrices S j are multiplied does not matterin the formula for the edge matrix E ( Z j ’s) but does matter in the formulas for the trianglematrices M left ( X i ’s) and M right ( X i ’s). Figure 7.
Edge matricesLastly, define the clockwise U-turn matrix U in SL n ( C ) by U = ( − n − ... +1 − ∈ SL n ( C ) . UANTUM TRACES 11
Then, we associate to a clockwise U-turn arc (resp. counterclockwise U-turn arc ) γ , as shownon the left hand (resp. right hand) side of Figure 8, the U-turn matrix U (resp. transpose U T of the U-turn matrix). We assume that γ has no kinks. Notice that U T = − U (resp.= U ) when n is even (resp. odd). (a) Clockwise U-turn (b)
Counterclockwise U-turn
Figure 8.
U-turn matrices1.7.
Computing the classical trace polynomial.
Let γ be an immersed oriented closedcurve in the surface S such that γ is transverse to the ideal triangulation λ . We want tocalculate the classical trace polynomial (cid:102) Tr( γ )( X ± /ni ) in T n ( λ ) = C [ X ± /n , X ± /n , . . . , X ± /nN ]associated to the immersed curve γ . The polynomial (cid:102) Tr( γ )( X ± /ni ) can be described as beingobtained from a “state-sum”, “local-to-global”, or “transfer matrices” construction.More precisely, as we travel along the curve γ according to its orientation, assume γ crosses edges E j k for k = 1 , , . . . , K in that order, and assume γ crosses triangles T i k for i k = 1 , , . . . , K in that order.As the curve γ crosses the edge E j k , moving out of the triangle T i k − into the triangle T i k ,this defines an edge-crossing arc γ j k ; see § E j k = E (( Z j k ) j (cid:48) ’s) ∈ SL n ( T n ( λ ))to be the associated edge matrix, where the ( Z j k ) j (cid:48) ’s are the j (cid:48) = 1 , . . . , n − E j k measured from right to left as seen from T i k − as usual.As γ traverses the triangle T i k between two edges E j k and E j k +1 , it does one of three things: • the curve γ turns left, ending on E j k +1 (cid:54) = E j k ; see Figure 5; • or γ turns right, ending on E j k +1 (cid:54) = E j k ; see Figure 6; • or γ does a U-turn, thereby returning to the same edge E j k +1 = E j k ; see Figure 8.We also keep track of the following winding information: for the first and second items above,the number of full turns t k to the right that the curve γ makes while traversing the triangle T i k ; and for the third item above, the number of half turns 2 t k + 1 to the right that thecurve γ makes before coming back to the same edge E j k . Note t k ∈ Z . We will see thatthe turning integer t k associated to the curve γ as it traverses the triangle T i k will only berelevant when the integer n is even. Now, let the ( X i k ) i (cid:48) ’s be the i (cid:48) = 1 , . . . , ( n − n − / T i k ; see Figure 5. Let γ (cid:48) be the curve obtainedby “pulling tight” the kinks of γ . (In particular, γ and γ (cid:48) are homotopic, but not, in general,regularly homotopic.) Corresponding to the three items above: • the curve γ (cid:48) turns left, defining a left-moving arc γ (cid:48) and an associated left matrix M (cid:48) i k = M left (( X i k ) i (cid:48) ’s), see § • or the curve γ (cid:48) turns right, defining a right-moving arc γ (cid:48) and an associated rightmatrix M (cid:48) i k = M right (( X i k ) i (cid:48) ’s), see § • or the curve γ (cid:48) does a clockwise or counterclockwise U-turn, thereby returning to thesame edge E j k +1 = E j k and defining a U-turn arc γ (cid:48) , see § γ (cid:48) is either left- or right-moving, put M i k = ( − ( n − t k M (cid:48) i k ∈ SL n ( T n ( λ )) , and in the third case, where γ (cid:48) is a U-turn, put M i k = ( − ( n − t k U ∈ SL n ( C ) , where U is the U-turn matrix defined in § § γ (cid:48) is associated to U (resp. U T = ( − n − U ) if γ (cid:48) travelsclockwise hence t k = 0 (resp. counterclockwise hence t k = − Definition 6.
The classical trace polynomial (cid:102)
Tr( γ )( X ± /ni ) ∈ T n ( λ ) associated to the im-mersed oriented closed curve γ is defined by (cid:102) Tr( γ )( X ± /ni ) = Tr ( E j M i E j M i · · · E j K M i K ) ∈ T n ( λ ) = C [ X ± /n , . . . , X ± /nN ]where on the right hand side we have taken the usual matrix trace. Note that this is inde-pendent of where we start along the curve γ , since the trace is invariant under conjugation.2. Fock-Goncharov quantum tori and quantum matrices
We define quantum versions of the classical polynomial algebra and the classical localmonodromy matrices, and relate them to the dual quantum group SL qn . Throughout, wechoose q ∈ C − { } and ω = q /n a n -root of q . We begin with some algebraic preliminaries.2.1. Quantum tori, matrix algebras, and the Weyl quantum ordering.
Quantum tori.
Let P be an integer N (cid:48) × N (cid:48) anti-symmetric “Poisson” matrix. Definition 7.
The quantum torus (with n -roots) T ω ( P ) associated to P is the quotient of thefree algebra C { X /n , X − /n , . . . , X /nN (cid:48) , X − /nN (cid:48) } in the indeterminates X ± /ni by the two-sidedideal generated by the relations X m i /ni X m j /nj = ω P ij m i m j X m j /nj X m i /ni ( m i , m j ∈ Z ) ,X m/ni X − m/ni = X − m/ni X m/ni = 1 ( m ∈ Z ) . Put X ± i = ( X ± /ni ) n . We refer to the X ± /ni as generators , and the X i as quantumcoordinates , or just coordinates . Define the subset of fractions Z /n = { m/n ; m ∈ Z } ⊆ Q . Written in terms of the coordinates X i and the fractions r ∈ Z /n , the relations above becomethe more palatable X r i i X r j j = q P ij r i r j X r j j X r i i ( r i , r j ∈ Z /n ) ,X ri X − ri = X − ri X ri = 1 ( r ∈ Z /n ) . UANTUM TRACES 13
Matrix algebras.
Definition 8.
Let A be a, possibly non-commutative, algebra, and let n (cid:48) be a positiveinteger. The matrix algebra with coefficients in A , denoted M n (cid:48) ( A ), is the complex vectorspace of n (cid:48) × n (cid:48) matrices, equipped with the usual “left-to-right” multiplicative structure.Namely, the product MN of two matrices M and N is defined entry-wise by( MN ) ij = n (cid:48) (cid:88) k =1 M ik N kj ∈ A (1 (cid:54) i, j (cid:54) n (cid:48) ) . Here, we use the usual convention that the entry M ij of a matrix M is the entry in the i -throw and j -th column. Note that, crucially, the order of M ik and N kj in the above equationmatters since these elements might not commute.2.1.3. Weyl quantum ordering. If T is a quantum torus, then there is a set function[ − ] : T ! T called the Weyl quantum ordering , defined on “un-ordered” monomials m = αX r i · · · X r k i k where α ∈ C , by the equation [ m ] = (cid:16) q − (cid:80) (cid:54) a
Let C (Θ n ) denote the set of cornervertices C (Θ n ) = { ( n, , , (0 , n, , (0 , , n ) } of the discrete triangle Θ n ; see § P : (Θ n − C (Θ n )) × (Θ n − C (Θ n )) ! {− , − , , , } using the quiver with vertex set Θ n − C (Θ n ) illustrated in Figure 9. The function P is definedby sending the ordered tuple ( v , v ) of vertices of Θ n − C (Θ n ) to 2 (resp. −
2) if there isa solid arrow pointing from v to v (resp. v to v ), to 1 (resp. −
1) if there is a dottedarrow pointing from v to v (resp. v to v ), and to 0 if there is no arrow connecting v and v . Note that all of the small downward-facing triangles are oriented clockwise, and allof the small upward-facing triangles are oriented counterclockwise. By labeling the verticesof Θ n − C (Θ n ) by their coordinates ( a, b, c ) we may think of the function P as a N × N anti-symmetric matrix P = ( P abc,a (cid:48) b (cid:48) c (cid:48) ) called the Poisson matrix associated to the quiver.Here, N = 3( n −
1) + ( n − n − /
2; see § Figure 9.
Quiver defining the Fock-Goncharov quantum torus
Definition 9.
Define the
Fock-Goncharov quantum torus T ωn ( T ) = C [ X ± /n , X ± /n , . . . , X ± /nN ] ω associated to the triangle T to be the quantum torus T ω ( P ) defined by the N × N Poissonmatrix P , with generators X ± /ni = X ± /nabc for all ( a, b, c ) ∈ Θ n − C (Θ n ). Note that when q = ω = 1 this indeed recovers the classical polynomial algebra T n ( T ) for T ; see § j = 1 , , . . . , n − Z ± /nj (resp. Z (cid:48)± /nj and Z (cid:48)(cid:48)± /nj )in place of X ± /nj n − j ) (resp. X ± /nj ( n − j )0 and X ± /n j ( n − j ) ); see Figure 10. So, triangle-coordinates willbe denoted X i = X abc for ( a, b, c ) ∈ Int(Θ n ) while edge-coordinates will be denoted Z j , Z (cid:48) j , Z (cid:48)(cid:48) j . Remark 10.
As we will see later on, we think of the Z -coordinates as quantizations of “half”of their corresponding classical edge-coordinates. Intuitively, this is because the “other half”of each coordinate “lives” in an adjacent triangle. When viewed inside an ideal triangulation λ , an edge E of λ “splits” this classical edge-coordinate into its two “quantum halves”.2.3. Quantum left and right matrices.
Weyl quantum ordering (continued).
Recall, given a quantum torus T with Poissonmatrix P , the definition of the Weyl ordering [ − ]; see § − ] : M n (cid:48) ( T ) −! M n (cid:48) ( T ) , [ M ] ij = [ M ij ] . We want to consider a slightly different function of matrix algebras.Observe that there is a multi-valued function C [ X ± /n , X ± /n , . . . , X ± /nN (cid:48) ] m.v. −! T that sends a “commuting polynomial” p ∈ C [ X ± /n , X ± /n , . . . , X ± /nN (cid:48) ] to the set of all “non-commuting polynomials” { P p } ⊆ T that can be formed from p by treating its variables asnon-commuting. More precisely, this multi-valued function is the inverse of the projection ofthe free algebra C { X ± /n , X ± /n , . . . , X ± /nN (cid:48) } onto C [ X ± /n , X ± /n , . . . , X ± /nN (cid:48) ] followed bythe projection of the free algebra C { X ± /n , X ± /n , . . . , X ± /nN (cid:48) } onto the quantum torus T .There is then induced a multi-valued functionM n (cid:48) ( C [ X ± /n , X ± /n , . . . , X ± /nN (cid:48) ]) m.v. −! M n (cid:48) ( T ) . UANTUM TRACES 15
By the symmetric property ( ∗ ) of the Weyl ordering, this multi-valued function of matrixalgebras induces a well-defined single-valued set function by the composition[ − ] : M n (cid:48) ( C [ X ± /n , X ± /n , . . . , X ± /nN (cid:48) ]) m.v. −! M n (cid:48) ( T ) [ − ] −! M n (cid:48) ( T ) . We wish to apply this function [ − ] of matrix algebras to the classical matrices of § n (cid:48) = n ), in the case where the quantum torus T is taken to be the Fock-Goncharovquantum torus T ωn ( T ) for a triangle T . In this setting, the above function [ − ] becomes[ − ] : M n ( T n ( T )) −! M n ( T ωn ( T )) . In practice, this is computed by the usual formula [ M ] ij = [ M ij ], where on the right handside the classical variables X i ∈ T n ( T ) appearing in the polynomial entry M ij ∈ T n ( T ) aretreated as their quantum counterparts X i ∈ T ωn ( T ).2.3.2. Quantum left and right matrices.
We begin with the classical situation for a triangle T . An extended left-moving arc γ is similar to a left-moving arc, from § T ; see Figure 10. We think ofan extended left-moving arc γ as the concatenation of “half” of an edge-crossing arc γ / together with a left-moving arc γ together with another half of an edge-crossing arc γ / , asindicated in Figure 10; compare Remark 10. We refer to these halves of edge-crossing arcsas half-edge-crossing arcs . Similarly, we define extended right-moving arcs γ .Defined exactly as in § left matrices M left ( X i ’s), right matrices M right ( X i ’s), and edge matrices E ( Z j ’s) in M n ( T n ( T )) associated to non -extended left-moving arcs (Figure 5), non -extended right-moving arcs (Figure 6), and half-edge-crossing arcs, respectively. Definition 11.
To an extended left-moving arc γ , as in Figure 10, we associate a quantumleft matrix L ω in M n ( T ωn ( T )) by the formula L ω = L ω ( X i , Z j , Z (cid:48) j ’s) = (cid:2) E ( Z j ’s) M left ( X i ’s) E ( Z (cid:48) j ’s) (cid:3) ∈ M n ( T ωn ( T ))where we have applied the Weyl quantum ordering [ − ] discussed in § E ( Z j ’s) M left ( X i ’s) E ( Z (cid:48) j ’s) of classical matrices in M n ( T n ( T )).Similarly, to an extended right-moving arc γ , as in Figure 10, we associate a quantum rightmatrix R ω in M n ( T ωn ( T )) by the formula R ω = R ω ( X i , Z j , Z (cid:48)(cid:48) j ’s) = (cid:2) E ( Z j ’s) M right ( X i ’s) E ( Z (cid:48)(cid:48) j ’s) (cid:3) ∈ M n ( T ωn ( T )) . Figure 10.
Quantum left and right matrices
Quantum SL n and its points: first theorem. Let A be a complex, possibly non-commutative, unital algebra. Definition 12.
We say that a 2 × M = ( a bc d ) in M ( A ) is an A -point of the quantummatrix algebra M q , denoted M ∈ M q ( A ) ⊆ M ( A ), if( ∗ ) ba = qab, dc = qcd, ca = qac, db = qbd, bc = cb, da − ad = ( q − q − ) bc ∈ A . We say that a matrix M ∈ M ( A ) is an A -point of the quantum special linear group SL q ,denoted M ∈ SL q ( A ) ⊆ M q ( A ) ⊆ M ( A ), if M ∈ M q ( A )and in addition the quantum determinantDet q ( M ) = da − qbc = ad − q − bc = 1 ∈ A is equal to 1.These notions are also defined for n × n matrices, as follows. Definition 13.
We say that a n × n matrix M ∈ M n ( A ) is an A -point of the quantum matrixalgebra M qn , denoted M ∈ M qn ( A ) ⊆ M n ( A ), if every 2 × M is an A -point ofM q . Equivalently, M im M ik = qM ik M im , M jm M im = qM im M jm ,M im M jk = M jk M im , M jm M ik − M ik M jm = ( q − q − ) M im M jk , for all i < j and k < m , where 1 (cid:54) i, j, k, m (cid:54) n .There also exists a notion of the quantum determinant Det q ( M ) ∈ A of a quantum matrix M ∈ M qn ( A ). We say that a matrix M ∈ M n ( A ) is an A -point of the quantum special lineargroup SL qn , denoted M ∈ SL qn ( A ) ⊆ M qn ( A ) ⊆ M n ( A ), if both M ∈ M qn ( A ) and the quantumdeterminant Det q ( M ) = 1.Observe that the subsets M qn ( A ) ⊆ M n ( A ) and SL qn ( A ) ⊆ M n ( A ) are not necessarily evenlinear subspaces (hence, nor subalgebras) of M n ( A ).Now, take A = T ωn ( T ) to be the Fock-Goncharov quantum torus for the triangle T , asdefined in § L ω and R ω in M n ( T ωn ( T )) be the quantum left and right matrices,respectively, as defined in Definition 11. In a companion paper [Dou], we prove: Theorem 14.
The quantum left and right matrices L ω , R ω ∈ SL qn ( T ωn ( T )) ⊆ M n ( T ωn ( T )) are T ωn ( T ) -points of the quantum special linear group SL qn . See the end of Remark 4. The proof uses a quantum version of the technology inventedby Fock and Goncharov called snakes; see Remark 5.2.5.
Example.
Consider the case n = 4; see Figure 11. On the right hand side of the figurewe show the commutation relations in the Fock-Goncharov quantum torus T ω ( T ), recallingFigure 9 and the definitions of § X Z (cid:48)(cid:48) = q X Z (cid:48)(cid:48) , X X = q − X X , Z Z = qZ Z , Z Z (cid:48) = q Z (cid:48) Z . UANTUM TRACES 17 (a)
Quantum left and right matrices (b)
Fock-Goncharov quiver
Figure 11.
Example in the case n = 4Then, the quantum left and right matrices are computed as L ω = (cid:34) Z − Z − Z − (cid:18) Z Z Z Z Z Z (cid:19) (cid:18) (cid:19) X − (cid:18) X (cid:19) X − (cid:18) X X (cid:19)(cid:18) (cid:19) X − (cid:18) X (cid:19) (cid:18) (cid:19) Z (cid:48)− Z (cid:48)− Z (cid:48)− (cid:32) Z (cid:48) Z (cid:48) Z (cid:48) Z (cid:48) Z (cid:48) Z (cid:48) (cid:33) (cid:35) and R ω = (cid:34) Z − Z − Z − (cid:18) Z Z Z Z Z Z (cid:19) (cid:18) (cid:19) X + (cid:18) X − (cid:19) X + (cid:32)
11 1 X − X − (cid:33)(cid:18) (cid:19) X + (cid:18) X − (cid:19) (cid:18) (cid:19) Z (cid:48)(cid:48)− Z (cid:48)(cid:48)− Z (cid:48)(cid:48)− (cid:32) Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48) (cid:33) (cid:35) . The theorem says that these two matrices are elements of SL q ( T ω ( T )). For instance, theentries a, b, c, d of the 2 × × L ω (cid:18) abcd (cid:19) = (cid:32) L ω L ω L ω L ω (cid:33) = [ Z Z Z Z (cid:48) Z (cid:48)− Z (cid:48)− X − X − X − ]+[ Z Z Z Z (cid:48) Z (cid:48)− Z (cid:48)− X − X X − ]++[ Z Z Z Z (cid:48) Z (cid:48)− Z (cid:48)− X X X − ][ Z Z Z Z (cid:48)− Z (cid:48)− Z (cid:48)− X − X − X − ][ Z Z Z − Z (cid:48) Z (cid:48)− Z (cid:48)− X − X − X − ]+[ Z Z Z − Z (cid:48) Z (cid:48)− Z (cid:48)− X − X X − ][ Z Z Z − Z (cid:48)− Z (cid:48)− Z (cid:48)− X − X − X − ] satisfy Equation ( ∗ ). For a computer verification of this, see Appendix A. We also demon-strate in the appendix that Equation ( ∗ ) is satisfied by the entries a, b, c, d of the 2 × × R ω (cid:18) abcd (cid:19) = (cid:32) R ω R ω R ω R ω (cid:33) = [ Z Z − Z − X X X Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48) ][ Z Z − Z − X X − X Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48)− ]+[ Z Z − Z − X X X Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48)− ][ Z − Z − Z − X X X Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48) ][ Z − Z − Z − X − X − X Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48)− ]+[ Z − Z − Z − X X − X Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48)− ]++[ Z − Z − Z − X X X Z (cid:48)(cid:48) Z (cid:48)(cid:48) Z (cid:48)(cid:48)− ] . Quantum tori for surfaces.
For a dotted ideal triangulation λ of S , in § T n ( λ ) = C [ X ± /n , X ± /n , . . . , X ± /nN ] where there isone generator X ± /ni associated to every dot on λ . In the case where S = T is an idealtriangle, in § T n ( T ) to a quantum torus T ωn ( T ). We now generalize the quantum torus T ωn ( T ) to a quantum torus T ωn ( λ ) associatedto the triangulated surface ( S , λ ) which deforms the classical polynomial algebra T n ( λ ).For each dotted triangle T of λ , associate a copy (cid:98) T of T , which is also a dotted triangle,such as that shown in Figure 3b. Notice that the boundary ∂ (cid:98) T consists of three ideal edges.The dotted ideal triangulation λ can be reconstructed from the individual triangles (cid:98) T bysupplying additional gluing data. To each dotted triangle (cid:98) T associate the Fock-Goncharovquantum torus T ωn ( (cid:98) T ) of the triangle (cid:98) T , whose coordinates we will denote by (cid:98) X .Recall that a generator (cid:98) X ± /nabc of the quantum torus T ωn ( (cid:98) T ) is either a triangle-generatoror an edge-generator. If (cid:98) X ± /nabc is an edge-generator, then there are two cases: • the corresponding generator X ± /ni in the classical polynomial algebra T n ( λ ) for theglued surface ( S , λ ) is a boundary-generator; • the corresponding classical generator X ± /ni in T n ( λ ) is an interior-generator.In the second case, the corresponding interior-generator X ± /ni in T n ( λ ) lies on an internaledge E of the ideal triangulation λ . So, there exists a triangle T (cid:48) adjacent to T along theedge E . Moreover, there exists a unique edge-generator (cid:98) X (cid:48)± /na (cid:48) b (cid:48) c (cid:48) in the quantum torus T ωn ( (cid:98) T (cid:48) )for the triangle (cid:98) T (cid:48) that also corresponds to the classical interior-generator X ± /ni in T n ( λ )lying on the internal edge E . Therefore, we may say that the two quantum generators (cid:98) X ± /nabc in T ωn ( (cid:98) T ) and (cid:98) X (cid:48)± /na (cid:48) b (cid:48) c (cid:48) in T ωn ( (cid:98) T (cid:48) ) correspond to one another ; see Figure 12. Definition 15.
The
Fock-Goncharov quantum torus T ωn ( λ ) associated to the surface S equipped with the dotted ideal triangulation λ is the sub-algebra T ωn ( λ ) ⊆ (cid:79) copies (cid:98) T of triangles T of λ T ωn ( (cid:98) T )of the tensor product of the Fock-Goncharov quantum tori T ωn ( (cid:98) T ) associated to the copies (cid:98) T of the dotted triangles T of the ideal triangulation λ , generated: • by triangle-generators (cid:98) X ± /nabc in T ωn ( (cid:98) T ); • by tensor products (cid:98) X ± /nabc ⊗ (cid:98) X (cid:48)± /na (cid:48) b (cid:48) c (cid:48) in T ωn ( (cid:98) T ) ⊗ T ωn ( (cid:98) T (cid:48) ) of corresponding edge-generatorsassociated to a common internal edge E lying at the interface between two triangles T and T (cid:48) in λ ; • and by edge-generators (cid:98) X ± /nabc in T ωn ( (cid:98) T ) associated to boundary edges E ⊆ ∂ S of theideal triangulation λ .In particular, when q = ω = 1, the Fock-Goncharov quantum torus T n ( λ ) is naturallyisomorphic to the classical polynomial algebra C [ X ± /n , X ± /n , . . . , X ± /nN ].From now on, we will omit the “hat” symbol in the notation, identifying the triangles T with their copies (cid:98) T . Remark 16.
A crucial difference between the local quantum tori T ωn ( T ) for the triangles T and the global quantum torus T ωn ( λ ) for the triangulated surface ( S , λ ) is that two edge-generators X ± /nabc and X ± /nABC in T ωn ( T ) lying on the same boundary edge of T may q -commute, UANTUM TRACES 19 but the corresponding interior-generators X ± /nabc ⊗ X (cid:48)± /na (cid:48) b (cid:48) c (cid:48) and X ± /nABC ⊗ X (cid:48)± /nA (cid:48) B (cid:48) C (cid:48) in T ωn ( λ ) alwayscommute. This is because the two triangles’ T and T (cid:48) quivers’ orientations “go against eachother”; see Figures 9 and 12. Intuitively, the local q -commutation relations on the boundaryare created by “splitting the edge-coordinates in half” at the quantum level; see Remarks 4and 10. This phenomenon does not occur for SL ( C ) because in that case each edge carriesonly one coordinate. (a) Before gluing (b)
After gluing
Figure 12.
Interior-generators as tensor products of local edge-generators3.
Main theorem: quantum trace polynomial
Framed oriented links in thickened surfaces.
So far, we have been working in the2-dimensional setting of the punctured surface S . From now on, we work in the 3-dimensionalsetting of the thickened surface S × (0 , § Definition 17. A framed oriented link K in the thickened surface S × (0 ,
1) is a compactoriented one-dimensional manifold, possibly-with-boundary, K ⊆ S × (0 ,
1) that is embeddedin S × (0 ,
1) and is equipped with a framing , satisfying the following properties:(1) we have ∂K = K ∩ (( ∂ S ) × (0 , K is vertical , meaning parallel to the (0 ,
1) axis,and points in the 1 direction;(3) for each boundary component k of S , the finitely many points K ∩ ( k × (0 , heights , meaning that the coordinates with respect to (0 ,
1) are distinct.Here, by a framing, we mean the choice of a continuous assignment along the link K of non-zero vectors in the tangent bundle of S × (0 ,
1) such that this vector field on K is everywheretransverse (that is, non-parallel) to K .A framed oriented knot K is a closed framed oriented link (namely, a framed oriented linkwith empty boundary ∂K = ∅ ) with one connected component.Two framed oriented links K and K (cid:48) are isotopic if K can be smoothly deformed to K (cid:48) through the class of framed oriented links.By possibly introducing kinks (Figures 18 and 19), one can always isotope a framed linkso that it has blackboard framing , meaning constant vertical framing in the 1 direction.It is common to think of a framed link K as a “ribbon”, namely a “thin” annulus embeddedin S × (0 ,
1) in which case we think of the framing as a continuous choice of vectors normalto the embedded annulus K . Remark 18.
Instead of using the picture conventions of [BW11b, § k × (0 ,
1) are higherand which are lower with respect to the (0 ,
1) direction. Note that by convention all linkdiagrams represent blackboard-framed links.3.2.
Stated links.
To deal algebraically with links-with-boundary, we need another concept.
Definition 19. A n - stated framed oriented link ( K, s ) is the data of a framed oriented link K equipped with a set function s : ∂K ! { , , . . . , n } called the state function , assigning to each element of the boundary of the link a state ,namely a number in { , , . . . , n } .As for links, there is the notion of isotopy of stated links.Notice that a stated closed link is the same thing as a closed link.3.3. Main theorem.
We now restrict to the case n = 3. Let the surface S be equipped witha dotted ideal triangulation λ . Recall the Fock-Goncharov quantum torus T ω ( λ ) associatedto this data; see Definition 15.Note that if K ⊆ S × (0 ,
1) is a framed oriented knot (meaning, in particular, that it isclosed), and if π : S × (0 , ! S × { } ∼ = S is the natural projection, then, possibly after anarbitrarily small perturbation of the knot K , we have that γ = π ( K ) is an immersed orientedclosed curve in S , and so we may consider the classical trace polynomial (cid:102) Tr( γ )( X ± / i ) in T ( λ ) = C [ X ± / , X ± / , . . . , X ± / N ] associated to γ ; see Definition 6.We now give a more detailed version of Theorem 2 from the Introduction. Theorem 20.
Let q ∈ C − { } be a non-zero complex number, and let ω = q / be a -rootof q . Then, there is a function of sets Tr ωλ : { isotopy classes of stated framed oriented links ( K, s ) in S × (0 , } −! T ω ( λ ) ⊆ (cid:79) triangles T of λ T ω ( T ) , sending a stated framed oriented link ( K, s ) in the thickened surface S × (0 , to a Laurentpolynomial Tr ωλ ( K, s ) = Tr ωλ ( K, s )( X ± / i ) ∈ T ω ( λ ) in the q -deformed (formal -roots of the) SL ( C ) -Fock-Goncharov coordinates X i , satisfying the following property: when q = ω = 1 and K = K (cid:116) K (cid:116) · · · (cid:116) K (cid:96) is a closed blackboard-framed oriented link thought of as adisjoint union of knots K j , then Tr λ ( K )( X ± / i ) = (cid:96) (cid:89) j =1 (cid:102) Tr( γ j )( X ± / i ) ∈ T ( λ ) = C [ X ± / , X ± / , . . . , X ± / N ] is the product of the corresponding classical trace polynomials, where γ j is the immersedoriented closed curve obtained by projecting the component K j of the closed link K to S .This construction also satisfies the HOMFLYPT skein relation for SL ( C ) , as well as theState Sum Property, which is the obvious generalization to the case n = 3 of Property (1) ofTheorem 11 in [BW11b, § . UANTUM TRACES 21
Remark 21.
A generalized version of Theorem 20 should hold in the case of SL n ( C ), byreplacing 3 with n everywhere in the statement. Below, we will in fact construct the quantumtrace Tr ωλ for general n , however we will only give a proof that it is well-defined in the case n = 3. When n = 2, our construction agrees with that of Bonahon and Wong. In particular,it gives a way to think of their construction, which was originally defined for un-orientedlinks, in terms of oriented links.4. Construction of SL n ( C ) -quantum traces As indicated in the previous remark, we will propose a definition for Tr ωλ for general n .Throughout, fix q ∈ C − { } and a n -root ω = q /n ∈ C − { } of q . For concreteness, alongthe way we will give explicit formulas for the case n = 3.4.1. Matrix conventions.
We will need to display 3 × × matrices. Lower indiceswill indicate rows and upper indices will indicate columns. A 3 × M = ( M ji ) willbe displayed in the general form M = (cid:18) M M M M M M M M M (cid:19) . A 3 × matrix M = ( M j j i i ) will be displayed in the general form M = M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M . If V and W are finite-dimensional complex vector spaces with bases { v , . . . , v m } and { w , . . . , w p } and if T : V ! W is a linear map, we define the p × m matrix [ T ] ∈ M p,m ( C )associated to T and these bases of V and W by the property T ( v j ) = p (cid:88) i =1 [ T ] ji w i ( j = 1 , , . . . , m ) . Biangles and the Reshetikhin-Turaev invariant. A biangle B is a closed disk withtwo punctures on its boundary. Biangles do not admit ideal triangulations, and so cannot bechosen for our general surface S . Nevertheless, we may still consider stated framed orientedlinks ( K, s ) in the thickened biangle B × (0 , C .In order to state the result, we need to define some elementary matrices associated tocertain local link diagrams, namely various U-turns and crossings.4.2.1. U-turns.
We begin with the U-turns. In Figures 13 and 14, we show the four possibleU-turns, which are in particular stated framed oriented links with the blackboard framing.In accordance with our picture conventions (see Remark 18), the boundary point of thelink that is labeled “Higher” or “H” is higher, namely has a greater coordinate with respectto the (0 ,
1) direction, than the boundary point of the link that is labeled “Lower” or “L”. (a)
Clockwise (b)
Counterclockwise
Figure 13.
Decreasing U-turns (a)
Counterclockwise (b)
Clockwise
Figure 14.
Increasing U-turns
Definition 22.
For general n , the SL n -coribbon element is ζ n = ( − n − q (1 − n ) /n = ( − n − ω n (1 − n ) ∈ C . The square root of the coribbon element is (cid:12)(cid:12)(cid:12)(cid:112) ζ n (cid:12)(cid:12)(cid:12) = + q − n n = + ω n (1 − n ) / ∈ C . Define a n × n matrix U q by U q = (cid:12)(cid:12)(cid:12)(cid:112) ζ n (cid:12)(cid:12)(cid:12) ( − n − q (1 − n ) / ... + q ( n − / − q ( n − / + q ( n − / ∈ M n ( C ) . Observe that the common ratio between adjacent entries in the matrix is equal to − q . Noticealso that when q = ω = 1 this recovers the classical U-turn matrix U from § n = 3, the 3 × U q is U q = (cid:18) q − / − q − / q − / (cid:19) ∈ M ( C ) . Now, for each pair of states s , s ∈ { , , . . . , n } , define four complex numbersTr ω B ( U cwdec ) s s , Tr ω B ( U ccwinc ) s s , Tr ω B ( U ccwdec ) s s , Tr ω B ( U cwinc ) s s ∈ C , UANTUM TRACES 23 by the matrix equations (cid:0) Tr ω B ( U cwdec ) s s (cid:1) = U q ∈ M n ( C ) , (cid:0) Tr ω B ( U ccwinc ) s s (cid:1) = ( U q ) T ∈ M n ( C ) , (cid:0) Tr ω B ( U ccwdec ) s s (cid:1) = ζ − n (cid:0) Tr ω B ( U cwdec ) s s (cid:1) ∈ M n ( C ) , (cid:0) Tr ω B ( U cwinc ) s s (cid:1) = ζ − n (cid:0) Tr ω B ( U ccwinc ) s s (cid:1) ∈ M n ( C );see § n = 3, in the above formulas ζ − = + q +8 / . Remark 23.
In the case n = 2, these formulas agree with those in [BW11b, Proposition13.2.b] for the underlying un-oriented link, taking α = − ω − and β = ω − ; compare [BW11b,Proposition 26].4.2.2. Crossings.
Next, we address crossings. Shown in Figures 15 and 16 are the eightpossible crossings, with blackboard framing and adhering to the usual picture conventions. (a) positive crossing,over strand higher tolower (b) negative crossing,over strand lower tohigher (c) positive crossing,over strand lower tohigher (d) negative crossing,over strand higher tolower
Figure 15.
Same direction crossings (a) negative crossing,over strand higher tolower (b) positive crossing,over strand lower tohigher (c) negative crossing,over strand lower tohigher (d) positive crossing,over strand higher tolower
Figure 16.
Opposite direction crossingsLet V be a n -dimensional complex vector space, and let V ∗ be the complex vector spacedual to V . Choose a linear basis { e , e , . . . , e n } for V , and let { e ∗ , e ∗ , . . . , e ∗ n } be the corre-sponding dual basis for V ∗ . Define four linear isomorphisms B V,V : V ⊗ V −! V ⊗ V, B V ∗ ,V ∗ : V ∗ ⊗ V ∗ −! V ∗ ⊗ V ∗ ,B V ∗ ,V : V ∗ ⊗ V −! V ⊗ V ∗ , B V,V ∗ : V ⊗ V ∗ −! V ∗ ⊗ V by extending linearly the following assignments for tensor product basis elements B V,V ( e i ⊗ e j ) = q +1 /n q − e i ⊗ e i , i = j, ( q − − q ) e i ⊗ e j + e j ⊗ e i , i < j,e j ⊗ e i , i > j,B V ∗ ,V ∗ ( e ∗ i ⊗ e ∗ j ) = q +1 /n q − e ∗ i ⊗ e ∗ i , i = j, ( q − − q ) e ∗ i ⊗ e ∗ j + e ∗ j ⊗ e ∗ i , i > j,e ∗ j ⊗ e ∗ i , i < j,B V ∗ ,V ( e ∗ i ⊗ e j ) = q − /n (cid:40) qe i ⊗ e ∗ i + ( q − q − ) (cid:80) (cid:54) k
We have the following equalities of matrices C q same = [ B V,V ] = [ B V ∗ ,V ∗ ] ∈ M n ( C ) , C q opp = [ B V ∗ ,V ] = [ B V,V ∗ ] ∈ M n ( C ) representing the linear isomorphisms B V,V , B V ∗ ,V ∗ , B V ∗ ,V and B V ∗ ,V when expressed in termsof the bases β V,V , β V ∗ ,V ∗ , β V ∗ ,V and β V,V ∗ . (cid:3) An observation is that, for general n , when q = ω = 1 then the two matrices C and C are identical. For another property of these so-called R -matrices , see § n = 3, these two 3 × matrices C q same and C q opp are given by C q same = q +1 / q − q − − q q − − q q − q − − q q − ∈ M ( C ) , C q opp = q +2 / q − q − q − q − − q q − q − − q q −
00 0 1 0 0 0 0 0 00 0 0 0 0 q − q − ∈ M ( C ) . UANTUM TRACES 25
Now, define for each quadruple of states s , s , s , s ∈ { , , . . . , n } eight complex numbersTr ω B ( C over-to-lowerpos-same ) s s s s , Tr ω B ( C over-to-higherneg-same ) s s s s , Tr ω B ( C over-to-higherpos-same ) s s s s , Tr ω B ( C over-to-lowerneg-same ) s s s s , Tr ω B ( C over-to-lowerneg-opp ) s s s s , Tr ω B ( C over-to-higherpos-opp ) s s s s , Tr ω B ( C over-to-higherneg-opp ) s s s s , Tr ω B ( C over-to-lowerpos-opp ) s s s s , by the matrix equations (cid:0) Tr ω B ( C over-to-lowerpos-same ) s s s s (cid:1) = C q same ∈ M n ( C ) , (cid:0) Tr ω B ( C over-to-higherneg-same ) s s s s (cid:1) = ( C q same ) − ∈ M n ( C ) , (cid:0) Tr ω B ( C over-to-higherpos-same ) s s s s (cid:1) = C q same ∈ M n ( C ) , (cid:0) Tr ω B ( C over-to-lowerneg-same ) s s s s (cid:1) = ( C q same ) − ∈ M n ( C ) , (cid:0) Tr ω B ( C over-to-lowerneg-opp ) s s s s (cid:1) = C q opp ∈ M n ( C ) , (cid:0) Tr ω B ( C over-to-higherpos-opp ) s s s s (cid:1) = ( C q opp ) − ∈ M n ( C ) , (cid:0) Tr ω B ( C over-to-higherneg-opp ) s s s s (cid:1) = C q opp ∈ M n ( C ) , (cid:0) Tr ω B ( C over-to-lowerpos-opp ) s s s s (cid:1) = ( C q opp ) − ∈ M n ( C ) . Remark 25.
In the case n = 2, these formulas agree with those in [BW11b, Lemma 22],for the underlying un-oriented link, taking A = ω − , α = − ω − and β = ω − (see [BW11b,Proposition 26]). In particular, as another reflection of the un-oriented nature of SL , when n = 2 the two matrices C q same and C q opp are identical for all q and ω (we saw above that, forgeneral n , this is only true for q = ω = 1). This can be explained conceptually as follows.For any n , the vector spaces V and V ∗ can be given the structure of a right SL qn -comodule .When n = 2, the linear isomorphism V ! V ∗ , x − qx ∗ , x x ∗ is an isomorphism ofright SL q -comodules, but this is not true for n >
2. Incidentally, this explains why, in aloose sense, the choices above for the bases β V,V , β V ∗ ,V ∗ , β V ∗ ,V and β V,V ∗ are “natural”.The linear isomorphisms B V,V , B V ∗ ,V ∗ , B V ∗ ,V and B V ∗ ,V arise naturally as braidings inthe ribbon category of finite-dimensional right SL qn -comodules, where the coribbon element ζ n is essentially given by Definition 22. Perhaps more interestingly, we have implicitly takena “symmetrized” duality, as opposed to the “usual” duality, which is more fitting to thecurrent 3-dimensional setting. (In the notation of [Kas95, Chapter XIV], our symmetrizeddualities b V and d V are related to the usual ones by b V = λb Kassel V and d V = λ − d Kassel V , where λ = q − n n = |√ ζ n | q ( n − / ; note that λ is the bottom left entry of U q , see § Trivial strand.
Lastly, consider a single strand crossing from one boundary edge ofthe biangle to the other boundary edge, as shown in Figure 17. Notice that the height ofthe strand with respect to the (0 ,
1) component does not play a role in this particular case.Then, this corresponds to the n × n identity matrix. Namely, define for each pair ofstates s , s ∈ { , , . . . , n } the complex number Tr ω B ( I ) s s by the matrix equation (cid:0) Tr ω B ( I ) s s (cid:1) = Id n ∈ M n ( C ) . Figure 17.
Single strand crossing the biangle4.2.4.
Kinks and the Reshetikhin-Turaev invariant for the biangle.
So far, we have shownhow to assign complex numbers to a handful of stated blackboard-framed oriented links inthe biangle B having one or two components.Now, let ( K, s ) be any stated blackboard-framed oriented link in the biangle having nokinks . By “combing” the link K into a bridge position, as described in [BW11b, §
4, proofof Lemma 15] but allowing for additional local pictures containing crossings, then we candefine a complex number Tr ω B ( K, s ) ∈ C by a State Sum Formula, which is identical to theformula appearing in [BW11b, §
4, p.1591].To define the complex number Tr ω B ( K, s ) ∈ C for an arbitrary stated framed orientedlink K in the thickened biangle B × (0 , K (cid:48) be a new stated blackboard-framedoriented link without kinks obtained by pulling tight the kinks in the un-framed oriented linkunderlying K . Then by above we have defined a complex number Tr ω B ( K (cid:48) , s ) ∈ C . Finally,define Tr ω B ( K, s ) by modifying Tr ω B ( K (cid:48) , s ) ∈ C according to the un-kinking “skein relations”shown in Figures 18 and 19. (For example, in the case n = 3, ζ = + q − / .) Note that thetwo positive (resp. negative) kinks shown are, in fact, locally regularly isotopic. Figure 18.
Skein relation for positive kinks
Figure 19.
Skein relation for negative kinks
UANTUM TRACES 27
Proposition 26 (Reshetikhin-Turaev invariant) . Let B be a biangle. The trace function Tr ω B : (cid:40) isotopy classes of stated framed oriented links K in B × (0 , (cid:41) −! C constructed in this section, sending a stated framed oriented link ( K, s ) in the thickenedbiangle B × (0 , to the complex number Tr ω B ( K, s ) ∈ C , is well-defined.This construction also satisfies the State Sum Property, which is the obvious generalizationto the case of general n of Property (1) of Proposition 13 in [BW11b, § . (cid:3) Remark 27.
The Reshetikhin-Turaev invariant can be defined more generally for ribbongraphs , also called webs . Bonahon and Wong define the SL ( C )-quantum trace by splittingthe edges of the ideal triangulation λ to form biangles and then “pushing all of the com-plexities of the link into the biangles,” [BW11b, p.1596] leaving only flat arcs lying over thetriangles. In order to construct the SL n ( C )-quantum trace for webs, one performs the sameprocedure, pushing all of the vertices of the web into the biangles. Then, the Reshetikhin-Turaev invariant can be applied to the webs in the biangles and (as we will see in the nextsection) the Fock-Goncharov matrices can be associated to the arcs lying over the triangles.4.3. Definition of the SL n ( C ) -quantum trace polynomial. Our construction of thequantum trace map in the case of general n will follow exactly the same procedure as ex-plained in [BW11b, §§ n = 2, where our Proposition 26 plays the role ofProposition 13 in [BW11b, § § §
6, p. 1600].Generalizing Property (2)(a) to the case of general n is accomplished by using the quantumleft and right matrices L ω and R ω , with coefficients in the Fock-Goncharov quantum torus T ωn ( T ) for a triangle T in the ideal triangulation λ , appearing in our Theorem 14. Considera single extended left-moving or right-moving arc crossing the triangle between two distinctboundary edges, such as that shown in Figure 10; see § n = 3, these extended left-moving and right-moving arcs areshown in Figures 20a and 20b. Using the notation from the figure, the quantum left andright matrices L ω and R ω are given by L ω = (cid:32) [ D − / L W Z XZ W ] [ D − / L W Z XW ]+[ D − / L W Z W ] [ D − / L W Z ]0 [ D − / L Z W ] [ D − / L Z ]0 0 [ D − / L ] (cid:33) where D − / L in T ω ( T ) is defined by D − / L = W − / Z − / X − / Z − / W − / , and R ω = (cid:32) [ D − / R W Z Z W ] 0 0[ D − / R Z Z W ] [ D − / R Z W ] 0[ D − / R Z W ] [ D − / R W ]+[ D − / R X − W ] [ D − / R X − ] (cid:33) where D − / R in T ω ( T ) is defined by D − / R = W − / Z − / X / Z − / W − / . This is the result of multiplying out the snake matrices in the case n = 3, analogous to the n = 4 examples that we gave in § (a) Left (b)
Right
Figure 20.
Quantum left and right matrices for n = 3 Remark 28.
In the above matrices, we recall that the square brackets surrounding eachmonomial indicate that we are taking the Weyl quantum ordering, which depends on thequiver defining the q -commutation relations in the Fock-Goncharov quantum torus T ωn ( T );see § L ω and R ω are points ofthe quantum special linear group SL qn . One puzzling observation is that, in order for thesematrices to satisfy even just the relations required to be in the quantum matrix algebra M qn ,they have to be normalized by “dividing out” their determinants. For example, the above n = 3 version of the matrix L ω would not satisfy the q -commutation relations required to bea point of M q if we had instead put D L = 1.Finally, generalizing Property (2)(a) of Theorem 11 in [BW11b, § s , s ∈ { , , . . . , n } two elements in the Fock-Goncharov quantum torus T ωn ( T ) for the triangle Tr ωλ ( L ) s s , Tr ωλ ( R ) s s ∈ T ωn ( T ) , by the matrix equations (cid:0) Tr ωλ ( L ) s s (cid:1) = L ω ∈ SL qn ( T ωn ( T )) ⊆ M n ( T ωn ( T )) , (cid:0) Tr ωλ ( R ) s s (cid:1) = R ω ∈ SL qn ( T ωn ( T )) ⊆ M n ( T ωn ( T ));see § n = 3, also serves as a reference for thecase of general n .This completes the definition of the SL n ( C )-quantum trace map Tr ωλ . One still has toprove it is well-defined, which requires harder work.Assuming well-definedness, note that the multiplicative property for the quantum tracemap, appearing at the bottom of Theorem 20 (and its generalized SL n ( C )-version), is provedexactly as in Lemma 19 and on p.1609 in [BW11b, §§
4, 6].Also, note that the quantum trace Tr ωλ ( K, s ) of a stated framed oriented link (
K, s ) canbe thought of as a tensor having dimension equal to the number of boundary points p i ∈ ∂K of the link, each associated to a state s i . If the states s i are arbitrarily partitioned into twogroups s i , . . . , s i (cid:96) and s j , . . . , s i m , then the quantum trace of the link can be written as amatrix (Tr ωλ ( K, s ) s j ,...,s jm s i ,...,s i(cid:96) ) with coefficients in T ωn ( λ ). UANTUM TRACES 29
HOMFLYPT skein relation.
One important skein relation is the well-known HOM-FLYPT relation from knot theory. It happens that the R -matrices for the SL n -quantumgroup satisfy this skein relation. This explains why these R -matrices can be used to recoverthe HOMFLYPT polynomial (and the Jones polynomial in the case n = 2).For us, this relation appears with the normalization displayed in Figure 21. One can checkfrom, say, Figures 15a, 15b, 17 together with the definitions of § q − /n C q same − q +1 /n ( C q same ) − = ( q − − q ) Id n ∈ M n ( C ) . Figure 21.
HOMFLYPT skein relation5.
Well-definedness: proof of the main theorem
To show the SL n ( C )-quantum trace map is well-defined, one has to check all of the orientedversions of the local moves depicted in Figures 15-19 in [BW11b, § n = 3, wedid this by hand, using a computer. Assuming this, the proof of Theorem 3.3 is complete.Here, and in Appendix A, we will demonstrate the kinds of algebraic calculations thatare needed in order to verify these local moves, thereby establishing that the quantum tracepolynomial of Theorem 20 is independent of ambient isotopy of the links K .The two most difficult moves are the (now oriented) moves of type (II) and (IV) appearingin [BW11b, §
5] in their Figures 16 and 18. Indeed, move (I) can be computed directly fromthe definitions, move (III) is essentially equivalent to Theorem 14, and move (V) followsfrom the kink-removing skein relations appearing in our Figures 18 and 19.
Figure 22.
One of the oriented versions of Move (II)In Figure 22, we show a representative oriented example of Move (II) in the case n = 3.Let K be the link on the left, and K (cid:48) the link on the right, as indicated. According to thedefinition of the quantum trace ( § × ∗ ) (cid:0) Tr ωλ ( K ) s s (cid:1) = (cid:16) A D E G H I (cid:17) (cid:18) q − / − q − / + q − / (cid:19) (cid:16) A D E G H I (cid:17) = = (cid:18) q − / A G q − / A H q − / A I − q − / E D + q − / D G − q − / E E + q − / D H q − / D I q − / I A − q − / H D + q − / G G − q − / H E + q − / G H q − / G I (cid:19) ? = (cid:16) a b c e f i (cid:17) = (cid:0) Tr ωλ ( K (cid:48) ) s s (cid:1) ∈ M ( T ω ( T )) , where we have used Figure 14a and the matrix (cid:0) Tr ω B ( U ccwinc ) s s (cid:1) = ( U q ) T from § L ω = L ω ( W , Z , W , Z , X ) andif we denote the right matrix by R ω = R ω ( W , Z , W , Z , X ) (see § (cid:16) A D E G H I (cid:17) = R ω ( W , Z , W , Z , X ) , (cid:16) A D E G H I (cid:17) = R ω ( W , Z , W , Z , X ) , (cid:16) a b c e f i (cid:17) = L ω ( W , Z , W , Z , X ) ∈ M ( T ω ( T )) . See Appendix A for a computer check of the above equality of 3 × ( T ω ( T ))representing this oriented Move (II) example. Figure 23.
One of the oriented versions of Move (IV)In Figure 23, we show a representative example of Move (IV) in the case n = 3. Let K bethe link on the left, and K (cid:48) the link on the right, as indicated. According to the definitionof the quantum trace ( § × matrices (see § ∗∗ ) (cid:0) Tr ωλ ( K ) s s s s (cid:1) = a A b A c A a D a E b D b E c D c E a G a H a I b G b H b I c G c H c I e A f A e D e E f D f E
00 0 0 e G e H e I f G f H f I i A i D i E
00 0 0 0 0 0 i G i H i I ? = q +1 / q − A a q − A b q − A c D a E a D b +( q − − q ) A e E b D c +( q − − q ) A f E c G a H a I a G b H b I b G c +( q − − q ) A i H c I c A e A f q − D e q − E e q − D f q − E f
00 0 0 G e H e I e G f +( q − − q ) D i H f +( q − − q ) E i I f A i D i E i
00 0 0 0 0 0 q − G i q − H i q − I i = (cid:0) Tr ωλ ( K (cid:48) ) s s s s (cid:1) ∈ M ( T ω ( T )) , where we have used Figure 15c and the matrix (cid:0) Tr ω B ( C over-to-higherpos-same ) s s s s (cid:1) = C q same from § UANTUM TRACES 31 left matrix by L ω = L ω ( W , Z , W , Z , X ) and if we denote the right matrix by R ω = R ω ( W , Z , W , Z , X ) (see § (cid:16) A D E G H I (cid:17) = R ω ( W , Z , W , Z , X ) , (cid:16) a b c e f i (cid:17) = L ω ( W , Z , W , Z , X ) . See Appendix A for a computer check of the above equality of 3 × matrices in M ( T ω ( T ))representing this oriented Move (IV) example.All of the oriented versions of all of the moves were similarly checked with the computer. Remark 29.
In the general case of SL n ( C ), a more conceptual proof of these same algebraicidentities (including Theorem 14), which are equivalent to the local isotopy moves discussedin this section, is given in [CS20] (motivated by [SS19, SS17]) in the context of quantumintegrable systems. Consequently, these works can be applied to finish the proof of thegeneralized version of Theorem 20 in the case of SL n ( C ), which is the main step towardproving Conjecture 1. References [AF17] N. Abdiel and C. Frohman. The localized skein algebra is Frobenius.
Algebr. Geom. Topol. ,17:3341–3373, 2017.[AK17] D. G. L. Allegretti and H. K. Kim. A duality map for quantum cluster varieties from surfaces.
Adv. Math. , 306:1164–1208, 2017.[All19] D. G. L. Allegretti. Categorified canonical bases and framed BPS states.
Selecta Math. (N.S.) ,25:69, 2019.[BFKB99] D. Bullock, C. Frohman, and J. Kania-Bartoszy´nska. Understanding the Kauffman bracket skeinmodule.
J. Knot Theory Ramifications , 8:265–277, 1999.[BG02] K. A. Brown and K. R. Goodearl.
Lectures on algebraic quantum groups . Birkh¨auser Verlag,Basel, 2002.[BHMV95] C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel. Topological quantum field theories derivedfrom the Kauffman bracket.
Topology , 34:883–927, 1995.[Bul97a] D. Bullock. Estimating a skein module with SL ( C ) characters. Proc. Amer. Math. Soc. ,125:1835–1839, 1997.[Bul97b] D. Bullock. Rings of SL ( C )-characters and the Kauffman bracket skein module. Comment.Math. Helv. , 72:521–542, 1997.[BW11a] F. Bonahon and H. Wong. Kauffman brackets, character varieties and triangulations of surfaces.
Contemp. Math. , 560:179–194, 2011.[BW11b] F. Bonahon and H. Wong. Quantum traces for representations of surface groups in SL ( C ). Geom. Topol. , 15:1569–1615, 2011.[BW16] F. Bonahon and H. Wong. Representations of the Kauffman bracket skein algebra I: invariantsand miraculous cancellations.
Invent. Math. , 204:195–243, 2016.[CF00] L. O. Chekhov and V. V. Fock. Observables in 3D gravity and geodesic algebras.
CzechoslovakJ. Phys. , 50:1201–1208, 2000.[CKKO20] S. Y. Cho, H. Kim, H. K. Kim, and D. Oh. Laurent positivity of quantized canonical bases forquantum cluster varieties from surfaces.
Comm. Math. Phys. , 373:655–705, 2020.[CKM14] S. Cautis, J. Kamnitzer, and S. Morrison. Webs and quantum skew Howe duality.
Math. Ann. ,360:351–390, 2014.[CL19] F. Costantino and T. T. Q. Lˆe. Stated skein algebras of surfaces. https://arxiv.org/abs/1907.11400 , 2019.[CM12] L. Charles and J. March´e. Multicurves and regular functions on the representation variety of asurface in SU(2).
Comment. Math. Helv. , 87:409–431, 2012.[Con70] J. H. Conway. An enumeration of knots and links, and some of their algebraic properties. In
Com-putational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) , pages 329–358. Pergamon,Oxford, 1970. [CS20] L. O. Chekhov and M. Shapiro. Darboux coordinates for symplectic groupoid and cluster alge-bras. https://arxiv.org/abs/2003.07499 , 2020.[CTT20] A. Casella, D. Tate, and S. Tillmann. Moduli spaces of real projective structures on surfaces. In
MSJ Memoirs , volume 38. Mathematical Society of Japan, Tokyo, 2020.[Dou] D. C. Douglas. Points of quantum SL n coming from quantum snakes. In preparation.[Dou20] D. C. Douglas. Classical and quantum traces coming from SL n ( C ) and U q ( sl n ). PhD thesis,University of Southern California, 2020.[DS20a] D. C. Douglas and Z. Sun. Tropical Fock-Goncharov coordinates for SL -webs on surfaces I:construction. https://arxiv.org/abs/2011.01768 , 2020.[DS20b] D. C. Douglas and Z. Sun. Tropical Fock-Goncharov coordinates for SL -webs on surfaces II:naturality. https://arxiv.org/abs/2012.14202 , 2020.[FC99] V. V. Fock and L. O. Chekhov. Quantum Teichm¨uller spaces. Teoret. Mat. Fiz. , 120:511–528,1999.[FG06a] V. V. Fock and A. B. Goncharov. Cluster X -varieties, amalgamation, and Poisson-Lie groups.In Algebraic geometry and number theory , volume 253 of
Progr. Math. , pages 27–68. Birkh¨auserBoston, Boston, MA, 2006.[FG06b] V. V. Fock and A. B. Goncharov. Moduli spaces of local systems and higher Teichm¨uller theory.
Publ. Math. Inst. Hautes ´Etudes Sci. , 103:1–211, 2006.[FG07a] V. V. Fock and A. B. Goncharov. Dual Teichm¨uller and lamination spaces.
Handbook of Te-ichm¨uller theory , 1:647–684, 2007.[FG07b] V. V. Fock and A. B. Goncharov. Moduli spaces of convex projective structures on surfaces.
Adv.Math. , 208:249–273, 2007.[FG09] V. V. Fock and A. B. Goncharov. Cluster ensembles, quantization and the dilogarithm.
Ann.Sci. ´Ec. Norm. Sup´er. , 42:865–930, 2009.[FKBL19] C. Frohman, J. Kania-Bartoszy´nska, and T. T. Q. Lˆe. Unicity for representations of the Kauffmanbracket skein algebra.
Invent. Math. , 215:609–650, 2019.[FKK13] B. Fontaine, J. Kamnitzer, and G. Kuperberg. Buildings, spiders, and geometric Satake.
Compos.Math. , 149:1871–1912, 2013.[Foc97] V. V. Fock. Dual Teichm¨uller spaces. https://arxiv.org/abs/dg-ga/9702018 , 1997.[Fon12] B. Fontaine. Generating basis webs for SL n . Adv. Math. , 229:2792–2817, 2012.[FS20] C. Frohman and A. S. Sikora. SU(3)-skein algebras and webs on surfaces. https://arxiv.org/abs/2002.08151 , 2020.[FWZ16] S. Fomin, L. Williams, and A. Zelevinsky. Introduction to Cluster Algebras. Chapters 1-3. https://arxiv.org/abs/1608.05735 , 2016.[FYH +
85] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. A new polynomialinvariant of knots and links.
Bull. Amer. Math. Soc. (N.S.) , 12:239–246, 1985.[FZ02] S. Fomin and A. Zelevinsky. Cluster algebras. I. Foundations.
J. Amer. Math. Soc. , 15:497–529,2002.[Gab17] M. Gabella. Quantum holonomies from spectral networks and framed BPS states.
Comm. Math.Phys. , 351:563–598, 2017.[GHKK18] M. Gross, P. Hacking, S. Keel, and M. Kontsevich. Canonical bases for cluster algebras.
J. Amer.Math. Soc. , 31:497–608, 2018.[GJS19a] I. Ganev, D. Jordan, and P. Safronov. The quantum Frobenius for character varieties and mul-tiplicative quiver varieties. https://arxiv.org/abs/1901.11450 , 2019.[GJS19b] S. Gunningham, D. Jordan, and P. Safronov. The finiteness conjecture for skein modules. https://arxiv.org/abs/1908.05233 , 2019.[GMN13] D. Gaiotto, G. W. Moore, and A. Neitzke. Spectral networks.
Ann. Henri Poincar´e , 14:1643–1731, 2013.[GMN14] D. Gaiotto, G. W. Moore, and A. Neitzke. Spectral networks and snakes.
Ann. Henri Poincar´e ,15:61–141, 2014.[Gol84] W. M. Goldman. The symplectic nature of fundamental groups of surfaces.
Adv. in Math. , 54:200–225, 1984.[Gol86] W. M. Goldman. Invariant functions on Lie groups and Hamiltonian flows of surface grouprepresentations.
Invent. Math. , 85:263–302, 1986.
UANTUM TRACES 33 [GS15] A. B. Goncharov and L. Shen. Geometry of canonical bases and mirror symmetry.
Invent. Math. ,202:487–633, 2015.[GS18] A. B. Goncharov and L. Shen. Donaldson-Thomas transformations of moduli spaces of G -localsystems. Adv. Math. , 327:225–348, 2018.[GS19] A. B. Goncharov and L. Shen. Quantum geometry of moduli spaces of local systems and repre-sentation theory. https://arxiv.org/abs/1904.10491 , 2019.[GSV09] M. Gekhtman, M. Shapiro, and A. Vainshtein. Poisson geometry of directed networks in a disk.
Selecta Math. (N.S.) , 15:61–103, 2009.[Hig20] V. Higgins. Triangular decomposition of SL skein algebras. https://arxiv.org/abs/2008.09419 , 2020.[Hit92] N. J. Hitchin. Lie groups and Teichm¨uller space. Topology , 31:449–473, 1992.[HN16] L. Hollands and A. Neitzke. Spectral networks and Fenchel-Nielsen coordinates.
Lett. Math.Phys. , 106:811–877, 2016.[HP93] J. Hoste and J. H. Przytycki. The (2 , ∞ )-skein module of lens spaces; a generalization of theJones polynomial. J. Knot Theory Ramifications , 2:321–333, 1993.[IY] T. Ishibashi and W. Yuasa. Skein and cluster algebras of marked surfaces without punctures for sl . In preparation.[Jae92] F. Jaeger. A new invariant of plane bipartite cubic graphs. Discrete Math. , 101:149–164, 1992.[Kas95] C. Kassel.
Quantum groups . Springer-Verlag, New York, 1995.[Kas98] R. M. Kashaev. Quantization of Teichm¨uller spaces and the quantum dilogarithm.
Lett. Math.Phys. , 43:105–115, 1998.[Kau87] L. H. Kauffman. State models and the Jones polynomial.
Topology , 26:395–407, 1987.[Kim20] H. K. Kim. A -laminations as basis for P GL cluster variety for surface. https://arxiv.org/abs/2011.14765 , 2020.[KLS20] H. K. Kim, T. T. Q. Lˆe, and M. Son. SL quantum trace in quantum Teichm¨uller theory viawrithe. https://arxiv.org/abs/1812.11628 , 2020.[KS97] A. Klimyk and K. Schm¨udgen. Quantum groups and their representations . Springer-Verlag,Berlin, 1997.[Kup96] G. Kuperberg. Spiders for rank 2 Lie algebras.
Comm. Math. Phys. , 180:109–151, 1996.[Lab06] F. Labourie. Anosov flows, surface groups and curves in projective space.
Invent. Math. , 165:51–114, 2006.[Lˆe] T. T. Q. Lˆe. The quantum trace for SL n skein algebra. In preparation.[Lˆe18] T. T. Q. Lˆe. Triangular decomposition of skein algebras. Quantum Topol. , 9:591–632, 2018.[Lˆe19] T. T. Q. Lˆe. Quantum Teichm¨uller spaces and quantum trace map.
J. Inst. Math. Jussieu ,18:249–291, 2019.[Lic93] W. B. R. Lickorish. The skein method for three-manifold invariants.
J. Knot Theory Ramifica-tions , 2:171–194, 1993.[LS] T. T. Q. Lˆe and A. S. Sikora. Private communication.[Maj95] S. Majid.
Foundations of quantum group theory . Cambridge University Press, Cambridge, 1995.[MFK94] D. Mumford, J. Fogarty, and F. Kirwan.
Geometric invariant theory. Third edition . Springer-Verlag, Berlin, 1994.[Mon93] S. Montgomery.
Hopf algebras and their actions on rings . American Mathematical Society, Prov-idence, RI, 1993.[MOY98] H. Murakami, T. Ohtsuki, and S. Yamada. Homfly polynomial via an invariant of colored planegraphs.
Enseign. Math. , 44:325–360, 1998.[NY20] A. Neitzke and F. Yan. q -nonabelianization for line defects. https://arxiv.org/abs/2002.08382 , 2020.[OY97] T. Ohtsuki and S. Yamada. Quantum SU(3) invariant of 3-manifolds via linear skein theory. J.Knot Theory Ramifications , 6:373–404, 1997.[Pro76] C. Procesi. The invariant theory of n × n matrices. Adv. Math. , 19:306–381, 1976.[Prz91] J. H. Przytycki. Skein modules of 3-manifolds.
Bull. Polish Acad. Sci. Math. , 39:91–100, 1991.[PS00] J. H. Przytycki and A. S. Sikora. On skein algebras and Sl ( C )-character varieties. Topology ,39:115–148, 2000. [PS19] J. H. Przytycki and A. S. Sikora. Skein algebras of surfaces.
Trans. Amer. Math. Soc. , 371:1309–1332, 2019.[PT87] J. H. Przytycki and P. Traczyk. Conway algebras and skein equivalence of links.
Proc. Amer.Math. Soc. , 100:744–748, 1987.[RT90] N. Y. Reshetikhin and V. G. Turaev. Ribbon graphs and their invariants derived from quantumgroups.
Comm. Math. Phys. , 127:1–26, 1990.[RT91] N. Y. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantumgroups.
Invent. Math. , 103:547–597, 1991.[Sik01] A. S. Sikora. SL n -character varieties as spaces of graphs. Trans. Amer. Math. Soc. , 353:2773–2804, 2001.[Sik05] A. S. Sikora. Skein theory for SU( n )-quantum invariants. Algebr. Geom. Topol. , 5:865–897, 2005.[SS17] G. Schrader and A. Shapiro. Continuous tensor categories from quantum groups I: algebraicaspects. https://arxiv.org/abs/1708.08107 , 2017.[SS19] G. Schrader and A. Shapiro. A cluster realization of U q ( sl n ) from quantum character varieties. Invent. Math. , 216:799–846, 2019.[Sun] Z. Sun. Tropical coordinates for SL n -webs on surfaces. In preparation.[SW07] A. S. Sikora and B. W. Westbury. Confluence theory for graphs. Algebr. Geom. Topol. , 7:439–478,2007.[Thu97] W. P. Thurston.
Three-dimensional geometry and topology. Vol. 1 . Princeton University Press,Princeton, NJ, 1997.[Thu14] D. P. Thurston. Positive basis for surface skein algebras.
Proc. Natl. Acad. Sci. USA , 111:9725–9732, 2014.[Tur88] V. G. Turaev. The Conway and Kauffman modules of a solid torus.
Zap. Nauchn. Sem. Leningrad.Otdel. Mat. Inst. Steklov. (LOMI) , 167:79–89, 190, 1988.[Tur89] V. G. Turaev. Algebras of loops on surfaces, algebras of knots, and quantization. In
Braid group,knot theory and statistical mechanics , pages 59–95. World Sci. Publ., Teaneck, NJ, 1989.[Tur91] V. G. Turaev. Skein quantization of Poisson algebras of loops on surfaces.
Ann. Sci. ´Ecole Norm.Sup. , 24:635–704, 1991.[Wit89] E. Witten. Quantum field theory and the Jones polynomial.
Comm. Math. Phys. , 121:351–399,1989.[Xie13] D. Xie. Higher laminations, webs and N = 2 line operators. https://arxiv.org/abs/1304.2390 , 2013. Department of Mathematics, Yale University, New Haven CT 06511, U.S.A.
Email address : [email protected] Appendix
A.In the following Mathematica code, Section 2 verifies the claims of § ∗ ) and ( ∗∗ ) in § ������ (* SECTION 1: DEFINITIONS NEEDED FOR QUANTUM CALCULATIONS *) (* number of coordinates *)(* the following definition of n is for Section 2;a different definition is given in Section 3 *) n = (* Poisson structure matrix *)(* Z3 = = = ′ = ′ = ′ = = = = ″ = ″ = ″ = *)(* Encodes, e.g., Z3 * Z2 = q^ ( ) Z2 * Z3, Z3 * Z3 ′ = q^ ( ) Z3 ′ * Z3, Z2 * Z3 = q^ (- ) Z3 * Z2 *)(* the following definition of P is for Section 2;a different definition is given in Section 3 *) P = {{
0, 1, 0, 2, 0, 0, 0, -
2, 0, 0, 0, 0 } , {-
1, 0, 1, 0, 0, 0, -
2, 2, 0, 0, 0, 0 } , { -
1, 0, 0, 0, 0, 2, 0, 0, 0, 0, - } , {-
2, 0, 0, 0, -
1, 0, 0, 2, 0, 0, 0, 0 } , {
0, 0, 0, 1, 0, -
1, 0, -
2, 2, 0, 0, 0 } , {
0, 0, 0, 0, 1, 0, 0, 0, -
2, 2, 0, 0 } , {
0, 2, -
2, 0, 0, 0, 0, -
2, 2, 0, -
2, 2 } , { -
2, 0, -
2, 2, 0, 2, 0, -
2, 0, 0, 0 } , {
0, 0, 0, 0, -
2, 2, -
2, 2, 0, -
2, 2, 0 } , {
0, 0, 0, 0, 0, -
2, 0, 0, 2, 0, -
1, 0 } , {
0, 0, 0, 0, 0, 0, 2, 0, -
2, 1, 0, - } , {
0, 0, 2, 0, 0, 0, -
2, 0, 0, 0, 1, 0 }} ; (* i [] : For encoding a monic monomial in the n variables *)(* E.g., Z3^ ( / )* Z2^ (- / )* Z1^ ( / )* Z3 is encoded by i [ / ] ,i [ - / ] ,i [ / ] ,i [ ] *) i [ z __] : = { z }[[ ]] + ⅈ { z }[[ ]] ; (* a [] : Weyl ordering [[ .,. ]] coefficient for a monomial *)(* E.g., since [[ Z3 * Z2 ]] = q^ (- / ) Z3 * Z2, then a i [ ] ,i [ ] = q^ (- / ) *) a [ w __] : = tempw = Flatten [ w ] ;temp = Sum Sum -( / ) * Im tempw i Im tempw j P Re tempw i , Re tempw j , j, i +
1, Length [ tempw ] , i, 1, Length [ tempw ] ; q ^ temp ; (* Test: *)(* a i [ ] ,i [ ] *)(* b [] : Coefficient resulting from re - ordering a monomial according to the order 1,2,...,n above *)(* E.g., since Z2 * Z3 = q^ (- ) Z3 * Z2, then b i [ ] ,i [ ] = q^ (- ) , but b i [ ] ,i [ ] = *) b [ w __] : = temp = = Flatten [ w ] ;Do Do If Re tempw Length [ tempw ] - i1 + ⩵ n - i3 + If Re tempw i2 ≥ n - i3 +
1, , temp = temp + * Im tempw Length [ tempw ] - i1 + Im tempw i2 P Re tempw Length [ tempw ] - i1 + , Re tempw i2 , i2, Length [ tempw ] - i1 +
2, Length [ tempw ] , i1, Length [ tempw ] , i3, n ; q ^ temp ; (* Test: *)(* b i [ ] ,i [ ] *)(* b i [ ] ,i [ ] *)(* b i [ ] ,i [ ] ,i [ ] *)(* c [] : Prints a possibly re - ordered monomial as if the variables commuted *)(* E.g., prints either Z3 * Z2 or Z2 * Z3 when Z2 * Z3 is entered, based on Mathematica's whims *)(* the following definition of c [] is for Section 2;a different definition is given in Section 3 *) c [ w __] : = temp = ConstantArray [
0, n ] ;tempw = Flatten [ w ] ;Do temp Re tempw i = temp Re tempw i + Im tempw i , i, Length [ tempw ] ; (* the following depends on the ordering of the n variables *)(* Z3 = = = ′ = ′ = ′ = = = = ″ = ″ = ″ = *) Z3 ^ temp [[ ]] * Z2 ^ temp [[ ]] * Z1 ^ temp [[ ]]* Z3 ′ ^temp [[ ]] * Z2 ′ ^ temp [[ ]] * Z1 ′ ^temp [[ ]] * X1 ^ temp [[ ]] * X2 ^ temp [[ ]] * X3 ^ temp [[ ]]* Z3 ″ ^ temp [[ ]] * Z2 ″ ^ temp [[ ]]* Z1 ″ ^ temp [[ ]] ; (* Test: *)* c i [ ] ,i [ ] *)(* c i [ ] ,i [ ] *)(* c i [ ] ,i [ ] ,i [ ] *)(* f [ m1,m2 ] : Computes [[ m _ ]]*[[ m _ ]] = q^r m,where m should be viewed as if in the order 1,2,...,n,but, as for c [] , Mathematica may not present the monomial in the proper order,so, for instance, if the output is q^ (- ) Z _ * Z _ (- ) Z _ * Z _ *) f [ x _ , y _] : = a [ x ] * a [ y ] * b [{ x, y }] * c [{ x, y }] ; (* Test: *)(* f i [ ] , i [ ] *)(* f i [ ] , i [ ] *)(* f i [ ] , i [ ] ,i [ ] *)(* g [ m ] , like f [ m1,m2 ] but for only one monomial m *)(* only used in Section 3 *) g [ x _] : = a [ x ] * b [ x ] * c [ x ] ; (* SECTION 2: QUANTUM LEFT AND RIGHT MATRICES *) (* SECTION 2.1:CHECKING COMMUTATION RELATIONS FOR QUANTIZED 2x2 SUB - MATRIX OF LEFT MATRIX *) (* Encoding 2x2 sub - matrix, e.g. a : = [[ a _ ]]+[[ a _ ]]+[[ a _ ]] *)(* Recall *)(* Z3 = = = ′ = ′ = ′ = = = = ″ = ″ = ″ = *) a1 = i [
1, 1 / ] , i [
2, 2 / ] , i [
3, 3 / ] , i [
4, 1 / ] , i [ - / ] ,i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [
10, 0 ] , i [
11, 0 ] , i [
12, 0 ] ;a2 = i [
1, 1 / ] , i [
2, 2 / ] , i [
3, 3 / ] , i [
4, 1 / ] , i [ - / ] , i [ - / ] ,i [ - / ] , i [
8, 2 / ] , i [ - / ] , i [
10, 0 ] , i [
11, 0 ] , i [
12, 0 ] ;a3 = i [
1, 1 / ] , i [
2, 2 / ] , i [
3, 3 / ] , i [
4, 1 / ] , i [ - / ] , i [ - / ] ,i [
7, 3 / ] , i [
8, 2 / ] , i [ - / ] , i [
10, 0 ] , i [
11, 0 ] , i [
12, 0 ] ;b1 = i [
1, 1 / ] , i [
2, 2 / ] , i [
3, 3 / ] , i [ - / ] , i [ - / ] , i [ - / ] ,i [ - / ] , i [ - / ] , i [ - / ] , i [
10, 0 ] , i [
11, 0 ] , i [
12, 0 ] ;c1 = i [
1, 1 / ] , i [
2, 2 / ] , i [ - / ] , i [
4, 1 / ] , i [ - / ] , i [ - / ] ,i [ - / ] , i [ - / ] , i [ - / ] , i [
10, 0 ] , i [
11, 0 ] , i [
12, 0 ] ;c2 = i [
1, 1 / ] , i [
2, 2 / ] , i [ - / ] , i [
4, 1 / ] , i [ - / ] , i [ - / ] ,i [ - / ] , i [
8, 2 / ] , i [ - / ] , i [
10, 0 ] , i [
11, 0 ] , i [
12, 0 ] ;d1 = i [
1, 1 / ] , i [
2, 2 / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ,i [ - / ] , i [ - / ] , i [ - / ] , i [
10, 0 ] , i [
11, 0 ] , i [
12, 0 ] ; �������� (* Checking relations *)(* da - ad = ( q - q^ (- )) bc *) Expand f d1, a1 + f d1, a2 + f d1, a3 - f a1, d1 + f a2, d1 + f a3, d1 Expand ( q - q ^ (- )) * f b1, c1 + f b1, c2 �������� - Z1 Z2 Z3q / X1 X3 Z1 ′ Z2 ′ Z3 ′ + q Z1 Z2 Z3X1 X3 Z1 ′ Z2 ′ Z3 ′ - q Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ + q / Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ �������� - Z1 Z2 Z3q / X1 X3 Z1 ′ Z2 ′ Z3 ′ + q Z1 Z2 Z3X1 X3 Z1 ′ Z2 ′ Z3 ′ - q Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ + q / Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ ��� appendix.nb ������� (* bc = cb *) Expand f b1, c1 + f b1, c2 Expand f c1, b1 + f c2, b1 �������� Z1 Z2 Z3q X1 X3 Z1 ′ Z2 ′ Z3 ′ + q / Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ �������� Z1 Z2 Z3q X1 X3 Z1 ′ Z2 ′ Z3 ′ + q / Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ �������� (* ca = qac *) Expand f [ c1, a1 ] + f [ c1, a2 ] + f [ c1, a3 ] + f [ c2, a1 ] + f [ c2, a2 ] + f [ c2, a3 ] Expand q * f [ a1, c1 ] + f [ a2, c1 ] + f [ a3, c1 ] + f [ a1, c2 ] + f [ a2, c2 ] + f [ a3, c2 ] �������� Z1 Z2 Z3 Z3 ′ X1 X3 Z1 ′ Z2 ′ + Z1 Z2 Z3 Z3 ′ q X1 X3 Z1 ′ Z2 ′ + X1 Z1 Z2 Z3 Z3 ′ X3 Z1 ′ Z2 ′ + q Z1 Z2 Z3 Z3 ′ X1 X2 X3 Z1 ′ Z2 ′ + X2 Z1 Z2 Z3 Z3 ′ q X1 X3 Z1 ′ Z2 ′ + X1 X2 Z1 Z2 Z3 Z3 ′ q X3 Z1 ′ Z2 ′ �������� Z1 Z2 Z3 Z3 ′ X1 X3 Z1 ′ Z2 ′ + Z1 Z2 Z3 Z3 ′ q X1 X3 Z1 ′ Z2 ′ + X1 Z1 Z2 Z3 Z3 ′ X3 Z1 ′ Z2 ′ + q Z1 Z2 Z3 Z3 ′ X1 X2 X3 Z1 ′ Z2 ′ + X2 Z1 Z2 Z3 Z3 ′ q X1 X3 Z1 ′ Z2 ′ + X1 X2 Z1 Z2 Z3 Z3 ′ q X3 Z1 ′ Z2 ′ �������� (* dc = acd *) Expand f d1, c1 + f d1, c2 Expand q * f c1, d1 + f c2, d1 �������� Z2 Z3X1 X3 Z1 Z1 ′ Z2 ′ Z3 ′ + q Z2 Z3X1 X2 X3 Z1 Z1 ′ Z2 ′ Z3 ′ �������� Z2 Z3X1 X3 Z1 Z1 ′ Z2 ′ Z3 ′ + q Z2 Z3X1 X2 X3 Z1 Z1 ′ Z2 ′ Z3 ′ �������� (* ba = qab *) Expand f b1, a1 + f b1, a2 + f b1, a3 Expand q * f a1, b1 + f a2, b1 + f a3, b1 �������� Z1 / Z2 Z3X1 X3 Z1 ′ Z2 ′ Z3 ′ + X1 Z1 / Z2 Z3X3 Z1 ′ Z2 ′ Z3 ′ + q Z1 / Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ �������� Z1 / Z2 Z3X1 X3 Z1 ′ Z2 ′ Z3 ′ + X1 Z1 / Z2 Z3X3 Z1 ′ Z2 ′ Z3 ′ + q Z1 / Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ �������� (* db = qdb *) Expand f d1, b1 Expand q * f b1, d1 �������� q Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ / �������� q Z1 Z2 Z3X1 X2 X3 Z1 ′ Z2 ′ Z3 ′ / appendix.nb ��� ������� (* SECTION 2.2:CHECKING COMMUTATION RELATIONS FOR QUANTIZED 2x2 SUB - MATRIX OF RIGHT MATRIX *) (* Encoding 2x2 sub - matrix, e.g. d : = d _ + d _ + d _ *)(* Recall *)(* Z3 = = = ′ = ′ = ′ = = = = ″ = ″ = ″ = *) a1 = i [
1, 1 / ] , i [ - / ] , i [ - / ] , i [
4, 0 ] , i [
5, 0 ] ,i [
6, 0 ] , i [
8, 1 / ] , i [
7, 1 / ] , i [
9, 1 / ] , i [
10, 1 / ] , i [
11, 1 / ] , i [
12, 3 / ] ;b1 = i [
1, 1 / ] , i [ - / ] , i [ - / ] , i [
4, 0 ] , i [
5, 0 ] , i [
6, 0 ] , i [
8, 1 / ] ,i [ - / ] , i [
9, 1 / ] , i [
10, 1 / ] , i [
11, 1 / ] , i [ - / ] ;b2 = i [
1, 1 / ] , i [ - / ] , i [ - / ] , i [
4, 0 ] , i [
5, 0 ] , i [
6, 0 ] , i [
8, 1 / ] ,i [
7, 1 / ] , i [
9, 1 / ] , i [
10, 1 / ] , i [
11, 1 / ] , i [ - / ] ;c1 = i [ - / ] , i [ - / ] , i [ - / ] , i [
4, 0 ] , i [
5, 0 ] , i [
6, 0 ] , i [
8, 1 / ] ,i [
7, 1 / ] , i [
9, 1 / ] , i [
10, 1 / ] , i [
11, 1 / ] , i [
12, 3 / ] ;d1 = i [ - / ] , i [ - / ] , i [ - / ] , i [
4, 0 ] , i [
5, 0 ] , i [
6, 0 ] , i [ - / ] ,i [ - / ] , i [
9, 1 / ] , i [
10, 1 / ] , i [
11, 1 / ] , i [ - / ] ;d2 = i [ - / ] , i [ - / ] , i [ - / ] , i [
4, 0 ] , i [
5, 0 ] , i [
6, 0 ] , i [
8, 1 / ] ,i [ - / ] , i [
9, 1 / ] , i [
10, 1 / ] , i [
11, 1 / ] , i [ - / ] ;d3 = i [ - / ] , i [ - / ] , i [ - / ] , i [
4, 0 ] , i [
5, 0 ] , i [
6, 0 ] , i [
8, 1 / ] ,i [
7, 1 / ] , i [
9, 1 / ] , i [
10, 1 / ] , i [
11, 1 / ] , i [ - / ] ; �������� (* da - ad = ( q - q^ (- )) bc *) Expand f d1, a1 + f d2, a1 + f d3, a1 - f a1, d1 + f a1, d2 + f a1, d3 Expand ( q - q ^ (- )) * f b1, c1 + f b2, c1 �������� - X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 + q X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 - X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 + q X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 �������� - X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 + q X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 - X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 + q X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 �������� (* bc = cb *) Expand f b1, c1 + f b2, c1 Expand f c1, b1 + f c1, b2 �������� X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 + X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 ��������
X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 + X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ Z1 Z2 Z3 �������� (* ca = qac *) Expand f [ c1, a1 ] Expand q * f [ a1, c1 ] �������� X1 X2 X3 Z1 ″ / Z2 ″ Z3 ″ q Z1 Z2 Z3 �������� X1 X2 X3 Z1 ″ / Z2 ″ Z3 ″ q Z1 Z2 Z3 ��� appendix.nb ������� (* dc = acd *) Expand f d1, c1 + f d2, c1 + f d3, c1 Expand q * f c1, d1 + f c1, d2 + f c1, d3 �������� X3 Z1 ″ Z2 ″ Z3 ″ q X2 Z1 Z2 Z3 / + X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 / + X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 / �������� X3 Z1 ″ Z2 ″ Z3 ″ q X2 Z1 Z2 Z3 / + X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 / + X1 X2 X3 Z1 ″ Z2 ″ Z3 ″ q Z1 Z2 Z3 / �������� (* ba = qab *) Expand f b1, a1 + f b2, a1 Expand q * f a1, b1 + f a1, b2 �������� q / X2 X3 Z1 ″ Z2 ″ Z3 Z3 ″ Z1 Z2 + q / X1 X2 X3 Z1 ″ Z2 ″ Z3 Z3 ″ Z1 Z2 �������� q / X2 X3 Z1 ″ Z2 ″ Z3 Z3 ″ Z1 Z2 + q / X1 X2 X3 Z1 ″ Z2 ″ Z3 Z3 ″ Z1 Z2 �������� (* db = qdb *) Expand f d1, b1 + f d2, b1 + f d3, b1 + f d1, b2 + f d2, b2 + f d3, b2 Expand q * f b1, d1 + f b1, d2 + f b1, d3 + f b2, d1 + f b2, d2 + f b2, d3 �������� X3 Z2 ″ Z3 ″ q X2 Z1 Z1 ″ Z2 Z3 + X3 Z2 ″ Z3 ″ q X1 X2 Z1 Z1 ″ Z2 Z3 + X2 X3 Z2 ″ Z3 ″ q Z1 Z1 ″ Z2 Z3 + q / X2 X3 Z2 ″ Z3 ″ Z1 Z1 ″ Z2 Z3 + X2 X3 Z2 ″ Z3 ″ q X1 Z1 Z1 ″ Z2 Z3 + q / X1 X2 X3 Z2 ″ Z3 ″ Z1 Z1 ″ Z2 Z3 ��������
X3 Z2 ″ Z3 ″ q X2 Z1 Z1 ″ Z2 Z3 + X3 Z2 ″ Z3 ″ q X1 X2 Z1 Z1 ″ Z2 Z3 + X2 X3 Z2 ″ Z3 ″ q Z1 Z1 ″ Z2 Z3 + q / X2 X3 Z2 ″ Z3 ″ Z1 Z1 ″ Z2 Z3 + X2 X3 Z2 ″ Z3 ″ q X1 Z1 Z1 ″ Z2 Z3 + q / X1 X2 X3 Z2 ″ Z3 ″ Z1 Z1 ″ Z2 Z3 (* SECTION 3: MOVES II AND IV EXAMPLES *) (* see Section 1 *) n = (* W1 = = = = = = =
7, going around in order clockwise *) P = {{ -
1, 0, 0, 0, -
2, 2 } , {
1, 0, 2, 0, 0, 0, - } , { -
2, 0, -
1, 0, 0, 2 } , {
0, 0, 1, 0, 2, 0, - } , {
0, 0, 0, -
2, 0, -
1, 2 } , {
2, 0, 0, 0, 1, 0, - } , {-
2, 2, -
2, 2, -
2, 2, 0 }} ;c [ w __] : = temp = ConstantArray [
0, n ] ;tempw = Flatten [ w ] ;Do temp Re tempw i = temp Re tempw i + Im tempw i , i, Length [ tempw ] ; (* W1 = = = = = = = *) W1 ^ temp [[ ]] * Z1 ^ temp [[ ]] * W2 ^ temp [[ ]]* Z2 ^temp [[ ]] * W3 ^ temp [[ ]] * Z3 ^temp [[ ]] * X ^ temp [[ ]] ; appendix.nb ��� ������� (* SECTION 3.1: ENCODING THE THREE LEFT MATRICES ( LOWER CASE LETTERS ) AND THREE RIGHT MATRICES ( CAPITAL LETTERS ) *) (* W1 = = = = = = = *)(* going from 1 - edge to 2 - edge *) a3 = i [
1, 2 / ] , i [
2, 1 / ] , i [
3, 1 / ] , i [
4, 2 / ] , i [
7, 2 / ] ;b31 = i [
1, 2 / ] , i [
2, 1 / ] , i [
3, 1 / ] , i [ - / ] , i [ - / ] ;b32 = i [
1, 2 / ] , i [
2, 1 / ] , i [
3, 1 / ] , i [ - / ] , i [
7, 2 / ] ;c3 = i [
1, 2 / ] , i [
2, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] ;e3 = i [ - / ] , i [
2, 1 / ] , i [
3, 1 / ] , i [ - / ] , i [ - / ] ;f3 = i [ - / ] , i [
2, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] ;i3 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ;A3 = i [
1, 1 / ] , i [
2, 2 / ] , i [
3, 2 / ] , i [
4, 1 / ] , i [
7, 1 / ] ;D3 = i [
1, 1 / ] , i [
2, 2 / ] , i [ - / ] , i [
4, 1 / ] , i [
7, 1 / ] ;E3 = i [
1, 1 / ] , i [ - / ] , i [ - / ] , i [
4, 1 / ] , i [
7, 1 / ] ;G3 = i [
1, 1 / ] , i [
2, 2 / ] , i [ - / ] , i [ - / ] , i [
7, 1 / ] ;H31 = i [
1, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ;H32 = i [
1, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [
7, 1 / ] ;I3 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ; (* going from 2 - edge to 3 - edge *) a1 = i [
3, 2 / ] , i [
4, 1 / ] , i [
5, 1 / ] , i [
6, 2 / ] , i [
7, 2 / ] ;b11 = i [
3, 2 / ] , i [
4, 1 / ] , i [
5, 1 / ] , i [ - / ] , i [ - / ] ;b12 = i [
3, 2 / ] , i [
4, 1 / ] , i [
5, 1 / ] , i [ - / ] , i [
7, 2 / ] ;c1 = i [
3, 2 / ] , i [
4, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] ;e1 = i [ - / ] , i [
4, 1 / ] , i [
5, 1 / ] , i [ - / ] , i [ - / ] ;f1 = i [ - / ] , i [
4, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] ;i1 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ;A1 = i [
3, 1 / ] , i [
4, 2 / ] , i [
5, 2 / ] , i [
6, 1 / ] , i [
7, 1 / ] ;D1 = i [
3, 1 / ] , i [
4, 2 / ] , i [ - / ] , i [
6, 1 / ] , i [
7, 1 / ] ;E1 = i [
3, 1 / ] , i [ - / ] , i [ - / ] , i [
6, 1 / ] , i [
7, 1 / ] ;G1 = i [
3, 1 / ] , i [
4, 2 / ] , i [ - / ] , i [ - / ] , i [
7, 1 / ] ;H11 = i [
3, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ;H12 = i [
3, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [
7, 1 / ] ;I1 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ; (* going from 3 - edge to 1 - edge *) a2 = i [
5, 2 / ] , i [
6, 1 / ] , i [
1, 1 / ] , i [
2, 2 / ] , i [
7, 2 / ] ;b21 = i [
5, 2 / ] , i [
6, 1 / ] , i [
1, 1 / ] , i [ - / ] , i [ - / ] ;b22 = i [
5, 2 / ] , i [
6, 1 / ] , i [
1, 1 / ] , i [ - / ] , i [
7, 2 / ] ;c2 = i [
5, 2 / ] , i [
6, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] ;e2 = i [ - / ] , i [
6, 1 / ] , i [
1, 1 / ] , i [ - / ] , i [ - / ] ;f2 = i [ - / ] , i [
6, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] ;i2 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ;A2 = i [
5, 1 / ] , i [
6, 2 / ] , i [
1, 2 / ] , i [
2, 1 / ] , i [
7, 1 / ] ;D2 = i [
5, 1 / ] , i [
6, 2 / ] , i [ - / ] , i [
2, 1 / ] , i [
7, 1 / ] ;E2 = i [
5, 1 / ] , i [ - / ] , i [ - / ] , i [
2, 1 / ] , i [
7, 1 / ] ;G2 = i [
5, 1 / ] , i [
6, 2 / ] , i [ - / ] , i [ - / ] , i [
7, 1 / ] ;H21 = i [
5, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ;H22 = i [
5, 1 / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [
7, 1 / ] ;I2 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ; �������� (* SECTION 3.2: CHECK OF MOVE ( II ) EXAMPLE *) (* *) Expand [( g [ a3 ])] Expand q ^ (- / ) * f [ A2, G1 ] �������� q / W1 / W2 / X / Z1 / Z2 / �������� q / W1 / W2 / X / Z1 / Z2 / ��� appendix.nb ������� (* *) Expand g b31 + g b32 Expand q ^ (- / ) * f [ A2, H11 ] + f [ A2, H12 ] �������� q / W1 / W2 / Z1 / X / Z2 / + W1 / W2 / X / Z1 / q / Z2 / �������� q / W1 / W2 / Z1 / X / Z2 / + W1 / W2 / X / Z1 / q / Z2 / �������� (* *) Expand [( g [ c3 ])] Expand q ^ (- / ) * f [ A2, I1 ] �������� q / W1 / Z1 / W2 / X / Z2 / �������� q / W1 / Z1 / W2 / X / Z2 / �������� (* *) Expand [( )] Expand - q ^ (- / ) * f [ E2, D1 ]+ q ^ (- / ) * f [ D2, G1 ] �������� �������� ��������� (* *) Expand [( g [ e3 ])] Expand - q ^ (- / ) * f [ E2, E1 ]+ q ^ (- / ) * f [ D2, H11 ] + f [ D2, H12 ] ��������� W2 / Z1 / q / W1 / X / Z2 / ��������� W2 / Z1 / q / W1 / X / Z2 / ��������� (* *) Expand g f3 Expand q ^ (- / ) * f [ D2, I1 ] ��������� Z1 / q / W1 / W2 / X / Z2 / ��������� Z1 / q / W1 / W2 / X / Z2 / ��������� (* *) Expand [( )] Expand q ^ (- / ) * f [ I2, A1 ]- q ^ (- / ) * f [ H21, D1 ] + f [ H22, D1 ] + q ^ (- / ) * f [ G2, G1 ] ��������� ��������� appendix.nb ��� �������� (* *) Expand [( )] Expand - q ^ (- / ) * f [ H21, E1 ] + f [ H22, E1 ] + q ^ (- / ) * f [ G2, H11 ] + f [ G2, H12 ] ��������� ��������� ��������� (* *) Expand g i3 Expand q ^ (- / ) * f [ G2, I1 ] ��������� / W1 / W2 / X / Z1 / Z2 / ��������� / W1 / W2 / X / Z1 / Z2 / (* SECTION 3.3: CHECK OF MOVE ( IV ) EXAMPLE *) (* / *) Expand f [ a3, A2 ] Expand q ^ ( / ) * q ^ (- ) * f [ A2, a3 ] ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ a3, D2 ] Expand q ^ ( / ) * f [ D2, a3 ] ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ a3, G2 ] Expand q ^ ( / ) * f [ G2, a3 ] ��������� q / W1 / W2 / W3 / X Z2 / Z3 / Z1 / ��������� q / W1 / W2 / W3 / X Z2 / Z3 / Z1 / ��������� (* / *) Expand f [ a3, E2 ] Expand q ^ ( / ) * f [ E2, a3 ] ��������� W1 / W2 / W3 / X Z1 / Z2 / q / Z3 / ��������� W1 / W2 / W3 / X Z1 / Z2 / q / Z3 / ��� appendix.nb �������� (* / *) Expand f [ a3, H21 ] + f [ a3, H22 ] Expand q ^ ( / ) * f [ H21, a3 ] + f [ H22, a3 ] ��������� W1 / W2 / W3 / Z2 / q / Z1 / Z3 / + W1 / W2 / W3 / X Z2 / q / Z1 / Z3 / ��������� W1 / W2 / W3 / Z2 / q / Z1 / Z3 / + W1 / W2 / W3 / X Z2 / q / Z1 / Z3 / ��������� (* / *) Expand f [ a3, I2 ] Expand q ^ ( / ) * f [ I2, a3 ] ��������� q / W1 / W2 / Z2 / W3 / Z1 / Z3 / ��������� q / W1 / W2 / Z2 / W3 / Z1 / Z3 / ��������� (* / *) Expand f b31, A2 + f b32, A2 Expand q ^ ( / ) * q ^ (- ) * f A2, b31 + f A2, b32 ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / + W1 / W2 / W3 / X Z1 / Z3 / q / Z2 / ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / + W1 / W2 / W3 / X Z1 / Z3 / q / Z2 / ��������� (* / *) Expand f b31, D2 + f b32, D2 Expand q ^ ( / ) * f D2, b31 + f D2, b32 + ( q^ (- ) - q ) * f [ A2, e3 ] ��������� W1 / W2 / W3 / Z1 / Z3 / q / Z2 / + q / W1 / W2 / W3 / X Z1 / Z3 / Z2 / ��������� W1 / W2 / W3 / Z1 / Z3 / q / Z2 / + q / W1 / W2 / W3 / X Z1 / Z3 / Z2 / ��������� (* / *) Expand f b31, G2 + f b32, G2 Expand q ^ ( / ) * f G2, b31 + f G2, b32 ��������� q / W1 / W2 / W3 / Z3 / Z1 / Z2 / + W1 / W2 / W3 / X Z3 / q / Z1 / Z2 / ��������� q / W1 / W2 / W3 / Z3 / Z1 / Z2 / + W1 / W2 / W3 / X Z3 / q / Z1 / Z2 / ��������� (* / *) Expand f [ e3, A2 ] Expand q ^ ( / ) * f [ A2, e3 ] ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / appendix.nb ��� �������� (* / *) Expand f [ e3, D2 ] Expand q ^ ( / ) * q ^ (- ) * f [ D2, e3 ] ��������� W2 / W3 / Z1 / Z3 / q / W1 / Z2 / ��������� W2 / W3 / Z1 / Z3 / q / W1 / Z2 / ��������� (* / *) Expand f [ e3, G2 ] Expand q ^ ( / ) * f [ G2, e3 ] ��������� q / W2 / W3 / Z3 / W1 / Z1 / Z2 / ��������� q / W2 / W3 / Z3 / W1 / Z1 / Z2 / ��������� (* / *) Expand f b31, E2 + f b32, E2 Expand q ^ ( / ) * f E2, b31 + f E2, b32 ��������� W1 / W2 / W3 / Z1 / q / Z2 / Z3 / + W1 / W2 / W3 / X Z1 / q / Z2 / Z3 / ��������� W1 / W2 / W3 / Z1 / q / Z2 / Z3 / + W1 / W2 / W3 / X Z1 / q / Z2 / Z3 / ��������� (* / *) Expand f b31, H21 + f b32, H21 + f b31, H22 + f b32, H22 Expand q ^ ( / ) * f H21, b31 + f H22, b31 + f H21, b32 + f H22, b32 ��������� W1 / W2 / W3 / q / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / + W1 / W2 / W3 / Xq / Z1 / Z2 / Z3 / ��������� W1 / W2 / W3 / q / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / + W1 / W2 / W3 / Xq / Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, E2 ] Expand q ^ ( / ) * q ^ (- ) * f [ E2, e3 ] ��������� W2 / W3 / Z1 / q / W1 / Z2 / Z3 / ��������� W2 / W3 / Z1 / q / W1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, H21 ] + f [ e3, H22 ] Expand q ^ ( / ) * f [ H21, e3 ] + f [ H22, e3 ] ��������� q / W2 / W3 / W1 / Z1 / Z2 / Z3 / + q / W2 / W3 / W1 / X Z1 / Z2 / Z3 / ��������� q / W2 / W3 / W1 / Z1 / Z2 / Z3 / + q / W2 / W3 / W1 / X Z1 / Z2 / Z3 / ������
7, 1 / ] ;I2 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ; �������� (* SECTION 3.2: CHECK OF MOVE ( II ) EXAMPLE *) (* *) Expand [( g [ a3 ])] Expand q ^ (- / ) * f [ A2, G1 ] �������� q / W1 / W2 / X / Z1 / Z2 / �������� q / W1 / W2 / X / Z1 / Z2 / ��� appendix.nb ������� (* *) Expand g b31 + g b32 Expand q ^ (- / ) * f [ A2, H11 ] + f [ A2, H12 ] �������� q / W1 / W2 / Z1 / X / Z2 / + W1 / W2 / X / Z1 / q / Z2 / �������� q / W1 / W2 / Z1 / X / Z2 / + W1 / W2 / X / Z1 / q / Z2 / �������� (* *) Expand [( g [ c3 ])] Expand q ^ (- / ) * f [ A2, I1 ] �������� q / W1 / Z1 / W2 / X / Z2 / �������� q / W1 / Z1 / W2 / X / Z2 / �������� (* *) Expand [( )] Expand - q ^ (- / ) * f [ E2, D1 ]+ q ^ (- / ) * f [ D2, G1 ] �������� �������� ��������� (* *) Expand [( g [ e3 ])] Expand - q ^ (- / ) * f [ E2, E1 ]+ q ^ (- / ) * f [ D2, H11 ] + f [ D2, H12 ] ��������� W2 / Z1 / q / W1 / X / Z2 / ��������� W2 / Z1 / q / W1 / X / Z2 / ��������� (* *) Expand g f3 Expand q ^ (- / ) * f [ D2, I1 ] ��������� Z1 / q / W1 / W2 / X / Z2 / ��������� Z1 / q / W1 / W2 / X / Z2 / ��������� (* *) Expand [( )] Expand q ^ (- / ) * f [ I2, A1 ]- q ^ (- / ) * f [ H21, D1 ] + f [ H22, D1 ] + q ^ (- / ) * f [ G2, G1 ] ��������� ��������� appendix.nb ��� �������� (* *) Expand [( )] Expand - q ^ (- / ) * f [ H21, E1 ] + f [ H22, E1 ] + q ^ (- / ) * f [ G2, H11 ] + f [ G2, H12 ] ��������� ��������� ��������� (* *) Expand g i3 Expand q ^ (- / ) * f [ G2, I1 ] ��������� / W1 / W2 / X / Z1 / Z2 / ��������� / W1 / W2 / X / Z1 / Z2 / (* SECTION 3.3: CHECK OF MOVE ( IV ) EXAMPLE *) (* / *) Expand f [ a3, A2 ] Expand q ^ ( / ) * q ^ (- ) * f [ A2, a3 ] ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ a3, D2 ] Expand q ^ ( / ) * f [ D2, a3 ] ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ a3, G2 ] Expand q ^ ( / ) * f [ G2, a3 ] ��������� q / W1 / W2 / W3 / X Z2 / Z3 / Z1 / ��������� q / W1 / W2 / W3 / X Z2 / Z3 / Z1 / ��������� (* / *) Expand f [ a3, E2 ] Expand q ^ ( / ) * f [ E2, a3 ] ��������� W1 / W2 / W3 / X Z1 / Z2 / q / Z3 / ��������� W1 / W2 / W3 / X Z1 / Z2 / q / Z3 / ��� appendix.nb �������� (* / *) Expand f [ a3, H21 ] + f [ a3, H22 ] Expand q ^ ( / ) * f [ H21, a3 ] + f [ H22, a3 ] ��������� W1 / W2 / W3 / Z2 / q / Z1 / Z3 / + W1 / W2 / W3 / X Z2 / q / Z1 / Z3 / ��������� W1 / W2 / W3 / Z2 / q / Z1 / Z3 / + W1 / W2 / W3 / X Z2 / q / Z1 / Z3 / ��������� (* / *) Expand f [ a3, I2 ] Expand q ^ ( / ) * f [ I2, a3 ] ��������� q / W1 / W2 / Z2 / W3 / Z1 / Z3 / ��������� q / W1 / W2 / Z2 / W3 / Z1 / Z3 / ��������� (* / *) Expand f b31, A2 + f b32, A2 Expand q ^ ( / ) * q ^ (- ) * f A2, b31 + f A2, b32 ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / + W1 / W2 / W3 / X Z1 / Z3 / q / Z2 / ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / + W1 / W2 / W3 / X Z1 / Z3 / q / Z2 / ��������� (* / *) Expand f b31, D2 + f b32, D2 Expand q ^ ( / ) * f D2, b31 + f D2, b32 + ( q^ (- ) - q ) * f [ A2, e3 ] ��������� W1 / W2 / W3 / Z1 / Z3 / q / Z2 / + q / W1 / W2 / W3 / X Z1 / Z3 / Z2 / ��������� W1 / W2 / W3 / Z1 / Z3 / q / Z2 / + q / W1 / W2 / W3 / X Z1 / Z3 / Z2 / ��������� (* / *) Expand f b31, G2 + f b32, G2 Expand q ^ ( / ) * f G2, b31 + f G2, b32 ��������� q / W1 / W2 / W3 / Z3 / Z1 / Z2 / + W1 / W2 / W3 / X Z3 / q / Z1 / Z2 / ��������� q / W1 / W2 / W3 / Z3 / Z1 / Z2 / + W1 / W2 / W3 / X Z3 / q / Z1 / Z2 / ��������� (* / *) Expand f [ e3, A2 ] Expand q ^ ( / ) * f [ A2, e3 ] ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / appendix.nb ��� �������� (* / *) Expand f [ e3, D2 ] Expand q ^ ( / ) * q ^ (- ) * f [ D2, e3 ] ��������� W2 / W3 / Z1 / Z3 / q / W1 / Z2 / ��������� W2 / W3 / Z1 / Z3 / q / W1 / Z2 / ��������� (* / *) Expand f [ e3, G2 ] Expand q ^ ( / ) * f [ G2, e3 ] ��������� q / W2 / W3 / Z3 / W1 / Z1 / Z2 / ��������� q / W2 / W3 / Z3 / W1 / Z1 / Z2 / ��������� (* / *) Expand f b31, E2 + f b32, E2 Expand q ^ ( / ) * f E2, b31 + f E2, b32 ��������� W1 / W2 / W3 / Z1 / q / Z2 / Z3 / + W1 / W2 / W3 / X Z1 / q / Z2 / Z3 / ��������� W1 / W2 / W3 / Z1 / q / Z2 / Z3 / + W1 / W2 / W3 / X Z1 / q / Z2 / Z3 / ��������� (* / *) Expand f b31, H21 + f b32, H21 + f b31, H22 + f b32, H22 Expand q ^ ( / ) * f H21, b31 + f H22, b31 + f H21, b32 + f H22, b32 ��������� W1 / W2 / W3 / q / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / + W1 / W2 / W3 / Xq / Z1 / Z2 / Z3 / ��������� W1 / W2 / W3 / q / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / + W1 / W2 / W3 / Xq / Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, E2 ] Expand q ^ ( / ) * q ^ (- ) * f [ E2, e3 ] ��������� W2 / W3 / Z1 / q / W1 / Z2 / Z3 / ��������� W2 / W3 / Z1 / q / W1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, H21 ] + f [ e3, H22 ] Expand q ^ ( / ) * f [ H21, e3 ] + f [ H22, e3 ] ��������� q / W2 / W3 / W1 / Z1 / Z2 / Z3 / + q / W2 / W3 / W1 / X Z1 / Z2 / Z3 / ��������� q / W2 / W3 / W1 / Z1 / Z2 / Z3 / + q / W2 / W3 / W1 / X Z1 / Z2 / Z3 / ������ appendix.nb �������� (* / *) Expand f b31, I2 + f b32, I2 Expand q ^ ( / ) * f I2, b31 + f I2, b32 ��������� W1 / W2 / q / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� W1 / W2 / q / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, I2 ] Expand q ^ ( / ) * f [ I2, e3 ] ��������� q / W2 / W1 / W3 / X Z1 / Z2 / Z3 / ��������� q / W2 / W1 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ c3, A2 ] Expand q ^ ( / ) * q ^ (- ) * f [ A2, c3 ] ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� (* / *) Expand f [ c3, D2 ] Expand q ^ ( / ) * f [ D2, c3 ] + ( q ^ (- ) - q ) * f A2, f3 ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� (* / *) Expand f [ c3, G2 ] Expand q ^ ( / ) * f [ G2, c3 ] + ( q ^ (- ) - q ) * f A2, i3 ��������� W1 / W3 / Z3 / q / W2 / Z1 / Z2 / ��������� W1 / W3 / Z3 / q / W2 / Z1 / Z2 / ��������� (* / *) Expand f f3, A2 Expand q ^ ( / ) * f A2, f3 ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / appendix.nb ��� �������� (* / *) Expand f f3, D2 Expand q ^ ( / ) * q ^ (- ) * f D2, f3 ��������� W3 / Z1 / Z3 / q / W1 / W2 / Z2 / ��������� W3 / Z1 / Z3 / q / W1 / W2 / Z2 / ��������� (* / *) Expand f f3, G2 Expand q ^ ( / ) * f G2, f3 + ( q ^ (- ) - q ) * f D2, i3 ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� (* / *) Expand f i3, A2 Expand q ^ ( / ) * f A2, i3 ��������� q / W1 / W3 / Z3 / W2 / Z1 / Z2 / ��������� q / W1 / W3 / Z3 / W2 / Z1 / Z2 / ��������� (* / *) Expand f i3, D2 Expand q ^ ( / ) * f D2, i3 ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� (* / *) Expand f i3, G2 Expand q ^ ( / ) * q ^ (- ) * f G2, i3 ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� (* / *) Expand f [ c3, E2 ] Expand q ^ ( / ) * f [ E2, c3 ] ��������� q / W1 / W3 / Z1 / W2 / Z2 / Z3 / ��������� q / W1 / W3 / Z1 / W2 / Z2 / Z3 / ������
7, 1 / ] ;I2 = i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] , i [ - / ] ; �������� (* SECTION 3.2: CHECK OF MOVE ( II ) EXAMPLE *) (* *) Expand [( g [ a3 ])] Expand q ^ (- / ) * f [ A2, G1 ] �������� q / W1 / W2 / X / Z1 / Z2 / �������� q / W1 / W2 / X / Z1 / Z2 / ��� appendix.nb ������� (* *) Expand g b31 + g b32 Expand q ^ (- / ) * f [ A2, H11 ] + f [ A2, H12 ] �������� q / W1 / W2 / Z1 / X / Z2 / + W1 / W2 / X / Z1 / q / Z2 / �������� q / W1 / W2 / Z1 / X / Z2 / + W1 / W2 / X / Z1 / q / Z2 / �������� (* *) Expand [( g [ c3 ])] Expand q ^ (- / ) * f [ A2, I1 ] �������� q / W1 / Z1 / W2 / X / Z2 / �������� q / W1 / Z1 / W2 / X / Z2 / �������� (* *) Expand [( )] Expand - q ^ (- / ) * f [ E2, D1 ]+ q ^ (- / ) * f [ D2, G1 ] �������� �������� ��������� (* *) Expand [( g [ e3 ])] Expand - q ^ (- / ) * f [ E2, E1 ]+ q ^ (- / ) * f [ D2, H11 ] + f [ D2, H12 ] ��������� W2 / Z1 / q / W1 / X / Z2 / ��������� W2 / Z1 / q / W1 / X / Z2 / ��������� (* *) Expand g f3 Expand q ^ (- / ) * f [ D2, I1 ] ��������� Z1 / q / W1 / W2 / X / Z2 / ��������� Z1 / q / W1 / W2 / X / Z2 / ��������� (* *) Expand [( )] Expand q ^ (- / ) * f [ I2, A1 ]- q ^ (- / ) * f [ H21, D1 ] + f [ H22, D1 ] + q ^ (- / ) * f [ G2, G1 ] ��������� ��������� appendix.nb ��� �������� (* *) Expand [( )] Expand - q ^ (- / ) * f [ H21, E1 ] + f [ H22, E1 ] + q ^ (- / ) * f [ G2, H11 ] + f [ G2, H12 ] ��������� ��������� ��������� (* *) Expand g i3 Expand q ^ (- / ) * f [ G2, I1 ] ��������� / W1 / W2 / X / Z1 / Z2 / ��������� / W1 / W2 / X / Z1 / Z2 / (* SECTION 3.3: CHECK OF MOVE ( IV ) EXAMPLE *) (* / *) Expand f [ a3, A2 ] Expand q ^ ( / ) * q ^ (- ) * f [ A2, a3 ] ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ a3, D2 ] Expand q ^ ( / ) * f [ D2, a3 ] ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ a3, G2 ] Expand q ^ ( / ) * f [ G2, a3 ] ��������� q / W1 / W2 / W3 / X Z2 / Z3 / Z1 / ��������� q / W1 / W2 / W3 / X Z2 / Z3 / Z1 / ��������� (* / *) Expand f [ a3, E2 ] Expand q ^ ( / ) * f [ E2, a3 ] ��������� W1 / W2 / W3 / X Z1 / Z2 / q / Z3 / ��������� W1 / W2 / W3 / X Z1 / Z2 / q / Z3 / ��� appendix.nb �������� (* / *) Expand f [ a3, H21 ] + f [ a3, H22 ] Expand q ^ ( / ) * f [ H21, a3 ] + f [ H22, a3 ] ��������� W1 / W2 / W3 / Z2 / q / Z1 / Z3 / + W1 / W2 / W3 / X Z2 / q / Z1 / Z3 / ��������� W1 / W2 / W3 / Z2 / q / Z1 / Z3 / + W1 / W2 / W3 / X Z2 / q / Z1 / Z3 / ��������� (* / *) Expand f [ a3, I2 ] Expand q ^ ( / ) * f [ I2, a3 ] ��������� q / W1 / W2 / Z2 / W3 / Z1 / Z3 / ��������� q / W1 / W2 / Z2 / W3 / Z1 / Z3 / ��������� (* / *) Expand f b31, A2 + f b32, A2 Expand q ^ ( / ) * q ^ (- ) * f A2, b31 + f A2, b32 ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / + W1 / W2 / W3 / X Z1 / Z3 / q / Z2 / ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / + W1 / W2 / W3 / X Z1 / Z3 / q / Z2 / ��������� (* / *) Expand f b31, D2 + f b32, D2 Expand q ^ ( / ) * f D2, b31 + f D2, b32 + ( q^ (- ) - q ) * f [ A2, e3 ] ��������� W1 / W2 / W3 / Z1 / Z3 / q / Z2 / + q / W1 / W2 / W3 / X Z1 / Z3 / Z2 / ��������� W1 / W2 / W3 / Z1 / Z3 / q / Z2 / + q / W1 / W2 / W3 / X Z1 / Z3 / Z2 / ��������� (* / *) Expand f b31, G2 + f b32, G2 Expand q ^ ( / ) * f G2, b31 + f G2, b32 ��������� q / W1 / W2 / W3 / Z3 / Z1 / Z2 / + W1 / W2 / W3 / X Z3 / q / Z1 / Z2 / ��������� q / W1 / W2 / W3 / Z3 / Z1 / Z2 / + W1 / W2 / W3 / X Z3 / q / Z1 / Z2 / ��������� (* / *) Expand f [ e3, A2 ] Expand q ^ ( / ) * f [ A2, e3 ] ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / ��������� q / W1 / W2 / W3 / Z1 / Z3 / Z2 / appendix.nb ��� �������� (* / *) Expand f [ e3, D2 ] Expand q ^ ( / ) * q ^ (- ) * f [ D2, e3 ] ��������� W2 / W3 / Z1 / Z3 / q / W1 / Z2 / ��������� W2 / W3 / Z1 / Z3 / q / W1 / Z2 / ��������� (* / *) Expand f [ e3, G2 ] Expand q ^ ( / ) * f [ G2, e3 ] ��������� q / W2 / W3 / Z3 / W1 / Z1 / Z2 / ��������� q / W2 / W3 / Z3 / W1 / Z1 / Z2 / ��������� (* / *) Expand f b31, E2 + f b32, E2 Expand q ^ ( / ) * f E2, b31 + f E2, b32 ��������� W1 / W2 / W3 / Z1 / q / Z2 / Z3 / + W1 / W2 / W3 / X Z1 / q / Z2 / Z3 / ��������� W1 / W2 / W3 / Z1 / q / Z2 / Z3 / + W1 / W2 / W3 / X Z1 / q / Z2 / Z3 / ��������� (* / *) Expand f b31, H21 + f b32, H21 + f b31, H22 + f b32, H22 Expand q ^ ( / ) * f H21, b31 + f H22, b31 + f H21, b32 + f H22, b32 ��������� W1 / W2 / W3 / q / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / + W1 / W2 / W3 / Xq / Z1 / Z2 / Z3 / ��������� W1 / W2 / W3 / q / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / + W1 / W2 / W3 / Xq / Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, E2 ] Expand q ^ ( / ) * q ^ (- ) * f [ E2, e3 ] ��������� W2 / W3 / Z1 / q / W1 / Z2 / Z3 / ��������� W2 / W3 / Z1 / q / W1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, H21 ] + f [ e3, H22 ] Expand q ^ ( / ) * f [ H21, e3 ] + f [ H22, e3 ] ��������� q / W2 / W3 / W1 / Z1 / Z2 / Z3 / + q / W2 / W3 / W1 / X Z1 / Z2 / Z3 / ��������� q / W2 / W3 / W1 / Z1 / Z2 / Z3 / + q / W2 / W3 / W1 / X Z1 / Z2 / Z3 / ������ appendix.nb �������� (* / *) Expand f b31, I2 + f b32, I2 Expand q ^ ( / ) * f I2, b31 + f I2, b32 ��������� W1 / W2 / q / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� W1 / W2 / q / W3 / Z1 / Z2 / Z3 / + q / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ e3, I2 ] Expand q ^ ( / ) * f [ I2, e3 ] ��������� q / W2 / W1 / W3 / X Z1 / Z2 / Z3 / ��������� q / W2 / W1 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ c3, A2 ] Expand q ^ ( / ) * q ^ (- ) * f [ A2, c3 ] ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� (* / *) Expand f [ c3, D2 ] Expand q ^ ( / ) * f [ D2, c3 ] + ( q ^ (- ) - q ) * f A2, f3 ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� (* / *) Expand f [ c3, G2 ] Expand q ^ ( / ) * f [ G2, c3 ] + ( q ^ (- ) - q ) * f A2, i3 ��������� W1 / W3 / Z3 / q / W2 / Z1 / Z2 / ��������� W1 / W3 / Z3 / q / W2 / Z1 / Z2 / ��������� (* / *) Expand f f3, A2 Expand q ^ ( / ) * f A2, f3 ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / ��������� q / W1 / W3 / Z1 / Z3 / W2 / Z2 / appendix.nb ��� �������� (* / *) Expand f f3, D2 Expand q ^ ( / ) * q ^ (- ) * f D2, f3 ��������� W3 / Z1 / Z3 / q / W1 / W2 / Z2 / ��������� W3 / Z1 / Z3 / q / W1 / W2 / Z2 / ��������� (* / *) Expand f f3, G2 Expand q ^ ( / ) * f G2, f3 + ( q ^ (- ) - q ) * f D2, i3 ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� (* / *) Expand f i3, A2 Expand q ^ ( / ) * f A2, i3 ��������� q / W1 / W3 / Z3 / W2 / Z1 / Z2 / ��������� q / W1 / W3 / Z3 / W2 / Z1 / Z2 / ��������� (* / *) Expand f i3, D2 Expand q ^ ( / ) * f D2, i3 ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� (* / *) Expand f i3, G2 Expand q ^ ( / ) * q ^ (- ) * f G2, i3 ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� W3 / Z3 / q / W1 / W2 / Z1 / Z2 / ��������� (* / *) Expand f [ c3, E2 ] Expand q ^ ( / ) * f [ E2, c3 ] ��������� q / W1 / W3 / Z1 / W2 / Z2 / Z3 / ��������� q / W1 / W3 / Z1 / W2 / Z2 / Z3 / ������ appendix.nb �������� (* / *) Expand f [ c3, H21 ] + f [ c3, H22 ] Expand q ^ ( / ) * f [ H21, c3 ] + f [ H22, c3 ] ��������� W1 / W3 / q / W2 / Z1 / Z2 / Z3 / + q / W1 / W3 / W2 / X Z1 / Z2 / Z3 / ��������� W1 / W3 / q / W2 / Z1 / Z2 / Z3 / + q / W1 / W3 / W2 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f f3, E2 Expand q ^ ( / ) * q ^ (- ) * f E2, f3 ��������� q / W3 / Z1 / W1 / W2 / Z2 / Z3 / ��������� q / W3 / Z1 / W1 / W2 / Z2 / Z3 / ��������� (* / *) Expand f f3, H21 + f f3, H22 Expand q ^ ( / ) * f H21, f3 + f H22, f3 + ( q^ (- ) - q ) * f E2, i3 ��������� W3 / q / W1 / W2 / Z1 / Z2 / Z3 / + q / W3 / W1 / W2 / X Z1 / Z2 / Z3 / ��������� W3 / q / W1 / W2 / Z1 / Z2 / Z3 / + q / W3 / W1 / W2 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f i3, E2 Expand q ^ ( / ) * f E2, i3 ��������� q / W3 / W1 / W2 / Z1 / Z2 / Z3 / ��������� q / W3 / W1 / W2 / Z1 / Z2 / Z3 / ��������� (* / *) Expand f i3, H21 + f i3, H22 Expand q ^ ( / ) * q ^ (- ) * f H21, i3 + f H22, i3 ��������� W3 / q / W1 / W2 / Z1 / Z2 / Z3 / + q / W3 / W1 / W2 / X Z1 / Z2 / Z3 / ��������� W3 / q / W1 / W2 / Z1 / Z2 / Z3 / + q / W3 / W1 / W2 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f [ c3, I2 ] Expand q ^ ( / ) * f [ I2, c3 ] ��������� W1 / q / W2 / W3 / X Z1 / Z2 / Z3 / ��������� W1 / q / W2 / W3 / X Z1 / Z2 / Z3 / appendix.nb ��� �������� (* / *) Expand f f3, I2 Expand q ^ ( / ) * f I2, f3 ��������� / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� (* / *) Expand f i3, I2 Expand q ^ ( / ) * q ^ (- ) * f I2, i3 ��������� / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ��������� / W1 / W2 / W3 / X Z1 / Z2 / Z3 / ������