HOMFLYPT skein sub-modules of the lens spaces L(p,1)
aa r X i v : . [ m a t h . G T ] M a y HOMFLYPT SKEIN SUB-MODULES OF THE LENS SPACES L ( p, IOANNIS DIAMANTIS
Abstract.
In this paper we work toward the HOMFLYPT skein module of L ( p, S ( L ( p, ′ , of the HOMFLYPT skein moduleof the solid torus ST, S (ST), which can be decomposed as the tensor product of the “positive”Λ ′ + and the “negative” Λ ′− sub-modules, and the Lambropoulou invariant, X , for knots andlinks in ST, that captures S (ST). It is a well-known result by now that S ( L ( p, S ( ST )
1) by imposing relations coming from theperformance of bbm’s and solving the infinite system of equations obtained that way.In this paper we work with a new basis of S (ST), Λ, and we relate the infinite system ofequations obtained by performing bbm’s on elements in Λ + to the infinite system of equationsobtained by performing bbm’s on elements in Λ − via a map I . More precisely we prove thatthe solutions of one system can be derived from the solutions of the other. Our aim is to reducethe complexity of the infinite system one needs to solve in order to compute S ( L ( p, Λ +
Skein modules were independently introduced by Przytycki [P] and Turaev [Tu] as general-izations of knot polynomials in S to knot polynomials in arbitrary 3-manifolds. The essence isthat skein modules are formal linear sums of (oriented) links in a 3-manifold M , modulo somelocal skein relations. Definition 1.
Let M be an oriented 3-manifold, R = Z [ u ± , z ± ], L the set of all oriented linksin M up to ambient isotopy in M and let S be the submodule of R L generated by the skeinexpressions u − L + − uL − − zL , where L + , L − and L comprise a Conway triple representedschematically by the illustrations in Figure 1. Figure 1.
The links L + , L − , L locally. Mathematics Subject Classification.
Key words and phrases.
HOMFLYPT polynomial, skein modules, solid torus, Iwahori–Hecke algebra of typeB, mixed links, mixed braids, lens spaces. igure 2. A basic element of S (ST).For convenience we allow the empty knot, ∅ , and add the relation u − ∅ − u ∅ = zT , where T denotes the trivial knot. Then the HOMFLYPT skein module of M is defined to be: S ( M ) = S (cid:0) M ; Z (cid:2) u ± , z ± (cid:3) , u − L + − uL − − zL (cid:1) = R L / S. The HOMFLYPT skein module of a 3-manifold is very hard to compute (see [HP] for a surveyon skein modules). For example, S ( S ) is freely generated by the unknot ([FYHLMO, PT]).Let now ST denote the solid torus. In [Tu], [HK] the Homflypt skein module of the solid torushas been computed using diagrammatic methods by means of the following theorem: Theorem 1 (Turaev, Kidwell–Hoste) . The skein module S (ST) is a free, infinitely generated Z [ u ± , z ± ] -module isomorphic to the symmetric tensor algebra SR b π , where b π denotes theconjugacy classes of non trivial elements of π (ST) . A basic element of S (ST) in the context of [Tu, HK], is illustrated in Figure 2. Note that inthe diagrammatic setting of [Tu] and [HK], ST is considered as Annulus × Interval. S (ST) is well-studied and understood by now. It forms a commutative algebra with multipli-cation induced by embedding two solid tori in one in a standard way. Let now B + denote thesub-algebra of S (ST), freely generated by elements that are clockwise oriented (see Fig. 2) andlet B − denote the sub-algebra of S (ST), freely generated by elements with counter-clockwiseorientation. Let also B + k denote the sub-module generated by elements in B + whose windingnumber is equal to k ∈ N and B −− k denote the sub-module generated by elements in B − whosewinding number is equal to k . As a linear space, B + is graded by B + ∼ = ⊕ k ≥ B + k and similarly, B − is graded by B − ∼ = ⊕ k ≥ B −− k . Finally, we have the following module decomposition: S (ST) = ⊕ λ,µ ≥ B −− λ ⊗ B + µ . The Turaev-basis of S (ST) is described in Equation 2 in open braid form (see left illustrationof Figure 5). In [DL2] a new basis, Λ, of S (ST) is presented via braids, that naturally describesisotopy in L ( p, B + and B − are presented in Equation 9and the sets B + k and B −− k are presented in Equations 10 and 11 respectively. n [DLP] the relation between S (ST) and S ( L ( p, S ( L ( p, S (ST) < bbm i > , where < bbm i > corresponds to the relations coming from the performance of all possible braidband moves (or slide moves) on elements in a basis of S (ST). More precisely, Eq. (1) suggeststhat in order to compute S ( L ( p, S (ST), apply all possiblebbm’s and identify all linear dependent elements. A step toward a simplification of the aboveinfinite system of equations can be found in [DL4], where it is shown that in order to compute S ( L ( p, aug (Equation 5,) and performbbm’s only on their first moving strand (the strand that lies closer to the surgery strand), i.e. S ( L ( p, Λ aug
1) ([DL1, LR1]) and in § S (ST). In § S (ST), which is crucial in order to obtain the new basis of S (ST), Λ, and in § ′ via an infinitetriangular matrix with invertible elements in the diagonal. Moreover, in § S ( L ( p, S (ST) presented in [DLP, DL4]. In § Λ aug +
Mixed Links in S . We consider ST to be the complement of a solid torus in S and knotsin ST are represented by mixed links in S . Mixed links consist of two parts, the unknottedfixed part b I that represents the complementary solid torus in S and the moving part L thatlinks with b I . A mixed link diagram is a diagram b I ∪ e L of b I ∪ L on the plane of b I , where this lane is equipped with the top-to-bottom direction of I (see top left hand side of Figure 3). Formore details on mixed links the reader is referred to [LR1, LR2, DL1] and references therein.The lens spaces L ( p,
1) can be obtained from S by surgery on the unknot with integer surgerycoefficient p . Surgery along the unknot can be realized by considering first the complementarysolid torus and then attaching to it a solid torus according to some homeomorphism on theboundary. Thus, isotopy in L ( p,
1) can be viewed as isotopy in ST together with the bandmoves in S , which reflect the surgery description of L ( p, α band moves (for an illustration see top of Figure 3) and thus, isotopy betweenoriented links in L ( p,
1) is reflected in S by means of the following result (cf. Thm. 5.8 [LR1],Thm. 6 [DL1] ): Two oriented links in L ( p, are isotopic if and only if two corresponding mixed link diagramsof theirs differ by isotopy in ST together with a finite sequence of the type α band moves. Mixed braids and braid equivalence for knots and links in L ( p, . By the Alexandertheorem for knots and links in the solid torus (cf. Thm. 1 [La2]), a mixed link diagram b I ∪ e L of b I ∪ L may be turned into a mixed braid I ∪ β with isotopic closure. This is a braid in S where,without loss of generality, its first strand represents b I , the fixed part, and the other strands, β ,represent the moving part L . The subbraid β is called the moving part of I ∪ β (see bottomleft hand side of Figure 3). Then, in order to translate isotopy for links in L ( p,
1) into braidequivalence, we first perform the technique of standard parting introduced in [LR2] in order toseparate the moving strands from the fixed strand that represents the lens spaces L ( p, over or under the fixed strand that lies on their right. Then, we define a braid band move to be amove between mixed braids, which is a band move between their closures. It starts with a littleband oriented downward, which, before sliding along a surgery strand, gets one twist positive or negative (see bottom of Figure 3).The sets of braids related to ST form groups, which are in fact the Artin braid groups of typeB, denoted B ,n , with presentation: B ,n = * t, σ , . . . , σ n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ tσ t = tσ tσ tσ i = σ i t, i > σ i σ i +1 σ i = σ i +1 σ i σ i +1 , ≤ i ≤ n − σ i σ j = σ j σ i , | i − j | > + , where the generators σ i and t are illustrated in Figure 4(i).Let now L denote the set of oriented knots and links in ST. Then, isotopy in L ( p,
1) is thentranslated on the level of mixed braids by means of the following theorem:
Theorem 2 (Theorem 5, [LR2]) . Let L , L be two oriented links in L ( p, and let I ∪ β , I ∪ β be two corresponding mixed braids in S . Then L is isotopic to L in L ( p, if and only if I ∪ β is equivalent to I ∪ β in B by the following moves: ( i ) Conjugation : α ∼ β − αβ, if α, β ∈ B ,n . ( ii ) Stabilization moves : α ∼ ασ ± n ∈ B ,n +1 , if α ∈ B ,n . ( iii ) Loop conjugation : α ∼ t ± αt ∓ , if α ∈ B ,n . ( iv ) Braid band moves : α ∼ t p α + σ ± , a + ∈ B ,n +1 , where α + is the word α with all indices shifted by +1. Note that moves (i), (ii) and (iii)correspond to link isotopy in ST . igure 3. Isotopy in L ( p,
1) and the two types of braid band moves on mixed braids.
Notation 1.
We denote a braid band move by bbm and, specifically, the result of a positive ornegative braid band move performed on the i th -moving strand of a mixed braid β by bbm ± i ( β ).Note also that in [LR2] it was shown that the choice of the position of connecting the twocomponents after the performance of a bbm is arbitrary.2. The HOMFLYPT skein module of ST via braids
In [La2] the most generic analogue of the HOMFLYPT polynomial, X , for links in the solidtorus ST has been derived from the generalized Iwahori–Hecke algebras of type B, H ,n , viaa unique Markov trace constructed on them. This algebra was defined as the quotient of C (cid:2) q ± (cid:3) B ,n over the quadratic relations g i = ( q − g i + q . Namely:H ,n ( q ) = C (cid:2) q ± (cid:3) B ,n h σ i − ( q − σ i − q i . It is also shown that the following sets form linear bases for H ,n ( q ) ([La2, Proposition 1 &Theorem 1]): ( i ) Σ n = { t k i . . . t k r i r · σ } , where 0 ≤ i < . . . < i r ≤ n − , ( ii ) Σ ′ n = { t ′ i k . . . t ′ i r k r · σ } , where 0 ≤ i < . . . < i r ≤ n − , where k , . . . , k r ∈ Z , t ′ = t := t, t ′ i = g i . . . g tg − . . . g − i and t i = g i . . . g tg . . . g i are the ‘looping elements’ in H ,n ( q ) (see Figure 4(ii)) and σ a basic element in the Iwahori–Heckealgebra of type A, H n ( q ), for example in the form of the elements in the set [Jo]: S n = (cid:8) ( g i g i − . . . g i − k )( g i g i − . . . g i − k ) . . . ( g i p g i p − . . . g i p − k p ) (cid:9) , igure 4. The generators of B ,n and the ‘looping’ elements t ′ i and t i .for 1 ≤ i < . . . < i p ≤ n − ′ n are used for constructing a Markov traceon H := S ∞ n =1 H ,n , and using this trace, a universal HOMFLYPT-type invariant for orientedlinks in ST was constructed. Theorem 3. [La2, Theorem 6 & Definition 1]
Given z, s k with k ∈ Z specified elements in R = C (cid:2) q ± (cid:3) , there exists a unique linear Markov trace function on H : tr : H → R ( z, s k ) , k ∈ Z determined by the rules: (1) tr( ab ) = tr( ba ) for a, b ∈ H ,n ( q )(2) tr(1) = 1 for all H ,n ( q )(3) tr( ag n ) = z tr( a ) for a ∈ H ,n ( q )(4) tr( at ′ nk ) = s k tr( a ) for a ∈ H ,n ( q ) , k ∈ Z Then, the function X : L → R ( z, s k ) X b α = ∆ n − · (cid:16) √ λ (cid:17) e tr ( π ( α )) , is an invariant of oriented links in ST , where ∆ := − − λq √ λ (1 − q ) , λ := z +1 − qqz , α ∈ B ,n is a wordin the σ i ’s and t ′ i ’s, b α is the closure of α , e is the exponent sum of the σ i ’s in α , π the canonicalmap of B ,n on H ,n ( q ) , such that t t and σ i g i . Remark 1.
Note that the use of the looping elements t ′ ’s enable the trace to be defined by justextending by rule (4) the three rules of the Ocneanu trace on the algebras H n ( q ) ([Jo]).In the braid setting of [La2], the elements of S (ST) correspond bijectively to the elements ofthe following set Λ ′ :(2) Λ ′ = { t k t ′ k . . . t ′ nk n , k i ∈ Z \ { } , k i ≤ k i +1 ∀ i, n ∈ N } . As explained in [La2, DL2], the set Λ ′ forms a basis of S (ST) in terms of braids (see also[HK, Tu]). Note that Λ ′ is a subset of H and, in particular, Λ ′ is a subset of Σ ′ = S n Σ ′ n .Note also that in contrast to elements in Σ ′ , the elements in Λ ′ have no gaps in the indices, theexponents are ordered and there are no ‘braiding tails’. Remark 2.
The Lambropoulou invariant X recovers S (ST). Indeed, it gives distinct values todistinct elements of Λ ′ , since tr( t k t ′ k . . . t ′ nk n ) = s k n . . . s k s k . igure 5. Elements in the two different bases of S (ST).2.1. A different basis for S (ST) . In [DL2], a different basis Λ for S (ST) is presented, whichis crucial toward the computation of S ( L ( p, Theorem 4. [DL2, Theorem 2]
The following set is a C [ q ± , z ± ] -basis for S (ST) : (3) Λ = { t k t k . . . t k n n , k i ∈ Z \ { } , k i ≤ k i +1 ∀ i, n ∈ N } . The importance of the new basis Λ of S (ST) lies in the simplicity of the algebraic expressionof a braid band move, which extends the link isotopy in ST to link isotopy in L ( p,
1) and thisfact was our motivation for establishing this new basis Λ. Note that comparing the set Λ withthe set Σ = S n Σ n , we observe that in Λ there are no gaps in the indices of the t i ’s and theexponents are in decreasing order. Also, there are no ‘braiding tails’ in the words in Λ.2.2. An ordering in the bases of S (ST) . We now define an ordering relation in the sets Σand Σ ′ , which passes to their respective subsets Λ and Λ ′ and that first appeared in [DL2]. Thisordering relation plays a crucial role to what will follow. For that we need the notion of the index of a word w in any of these sets, denoted ind ( w ). In Λ ′ or Λ ind ( w ) is defined to be thehighest index of the t ′ i ’s, resp. of the t i ’s in w . Similarly, in Σ ′ or Σ, ind ( w ) is defined as aboveby ignoring possible gaps in the indices of the looping generators and by ignoring the braidingparts in the algebras H n ( q ). Moreover, the index of a monomial in H n ( q ) is equal to 0. Definition 2. [DL2, Definition 2] Let w = t ′ i k . . . t ′ i µ k µ · β and u = t ′ j λ . . . t ′ j ν λ ν · β in Σ ′ ,where k t , λ s ∈ Z for all t, s and β , β ∈ H n ( q ). Then, we define the following ordering in Σ ′ :(a) If P µi =0 k i < P νi =0 λ i , then w < u .(b) If P µi =0 k i = P νi =0 λ i , then:(i) if ind ( w ) < ind ( u ), then w < u ,(ii) if ind ( w ) = ind ( u ), then:( α ) if i = j , . . . , i s − = j s − , i s < j s , then w > u ,( β ) if i t = j t for all t and k µ = λ µ , k µ − = λ µ − , . . . , k i +1 = λ i +1 , | k i | < | λ i | , then w < u ,( γ ) if i t = j t for all t and k µ = λ µ , k µ − = λ µ − , . . . , k i +1 = λ i +1 , | k i | = | λ i | and k i > λ i , then w < u , δ ) if i t = j t ∀ t and k i = λ i , ∀ i , then w = u .The ordering in the set Σ is defined as in Σ ′ , where t ′ i ’s are replaced by t i ’s. Notation 2.
We set τ k i,i + m i,i + m := t k i i . . . t k i + m i + m and τ ′ k i,i + m i,i + m := t ′ k i i . . . t ′ k i + m i + m , for m ∈ N , k j = 0 forall j .The subsets of level k , Λ ( k ) and Λ ′ ( k ) , of Λ and Λ ′ respectively ([DL2, Definition 3]), are definedto be the sets:(4) Λ ( k ) := { t k t k . . . t k m m | P mi =0 k i = k, k i ∈ Z \ { } , k i ≤ k i +1 ∀ i } Λ ′ ( k ) := { t ′ k t ′ k . . . t ′ mk m | P mi =0 k i = k, k i ∈ Z \ { } , k i ≤ k i +1 ∀ i } In [DL2] it was shown that the sets Λ ( k ) and Λ ′ ( k ) are totally ordered and well ordered for all k ([DL2, Propositions 1 & 2]). Note that in [DLP] the exponents in the monomials of Λ are indecreasing order, while here the exponents are considered in increasing order, which is totallysymmetric.We finally define the set Λ aug , which augments the basis Λ and its subset of level k , and wealso introduce the notion of homologous words . Definition 3.
We define the set:(5) Λ aug := { t k t k . . . t k n n , k i ∈ Z \{ }} . and the subset of level k , Λ aug ( k ) , of Λ aug :(6) Λ aug ( k ) := { t k t k . . . t k m m | m X i =0 k i = k, k i ∈ Z \{ }} Definition 4.
We shall say that two words w ′ ∈ Λ ′ and w ∈ Λ are homologous , denoted w ′ ∼ w ,if w is obtained from w ′ by turning t ′ i into t i for all i .2.3. Relating Λ ′ to Λ . We now present results from [DL2] used in order to relate the setsΛ ′ and Λ via a lower triangular matrix with invertible elements in the diagonal. We start byexpressing elements in Λ ′ to to expressions containing the t i ’s. We have that: Theorem 5 (Theorem 7, [DL2]) . The following relations hold in H ,n ( q ) for k ∈ Z : t k t ′ k . . . t ′ mk m = q − m P n =1 nk n · t k t k . . . t k m m + P i f i ( q ) · t k t k . . . t k m m · w i ++ P j g j ( q ) τ j · u j , where w i , u j ∈ H m +1 ( q ) , ∀ i , τ j ∈ Σ n , such that τ j < t k t k . . . t k m m , ∀ j . Equivalently, the relation in Theorem 5 can be written as:(7) t k t k . . . t k m m = q m P n =1 nk n · t k t ′ k . . . t ′ mk m + P i f ′ i ( q ) · t k t k . . . t k m m · w i ++ P j g ′ j ( q ) τ j · u j , hen applying Theorem 5 on an element in Λ ′ , we obtain the homologous word, the homol-ogous word followed by a braiding “tail”, and a sum of lower order terms followed by braiding“tails”. These elements belong to Σ n since they may have gaps in their indices, and we managethe gaps applying Theorem 8 in [DL2], namely:(8) Σ n ∋ τ b = X i f i ( q ) τ i · w i : τ i ∈ Λ aug w i ∈ H n ( q ) , ∀ i, where b = denotes that conjugation is applied in this process.We now deal with the elements in Λ aug ( k ) that are followed by a braiding “tail” w in H n ( q ).More precisely we have: Theorem 6 (Theorem 9, [DL2]) . For an element in Λ aug ( k ) followed by a braiding “tail” w in H n ( q ) we have that: tr ( τ · w ) = X j f j ( q, z ) · tr ( τ j ) , such that τ j ∈ Λ aug ( k ) and τ j < τ , for all j . One very important result in [DL2] is that one can change the order of the exponents by usingconjugation and stabilization moves on elements in Λ and express them as sums of monomialsin t i ’s with arbitrary exponents and which are of lower order than the initial elements in Λ.Note that both conjugation and stabilization moves are captures by the trace rules, and that wetranslate here Theorem 9 in [DL2] using the trace. Theorem 7. [DL2, Theorem 9]
For an element in Λ aug followed by a braiding “tail” in H n ( q ) we have that: tr ( τ k ,m ,m · w ) = tr X j τ λ ,j ,j · w j , where τ λ ,j ,j ∈ Λ and w, w j ∈ S n ∈ N H n ( q ) for all j . Examples of how to apply Theorems 5, 6 and 7 can be found in [DL3].2.4.
Relating S ( L ( p, to S (ST) . In order to simplify this system of equations (1), in [DLP]we first show that performing a bbm on a mixed braid in B ,n reduces to performing bbm’s onelements in the canonical basis, Σ ′ n , of the algebra H ,n ( q ) and, in fact, on their first movingstrand. We then reduce the equations obtained from elements in Σ ′ to equations obtained fromelements in Σ. In order now to reduce further the computation to elements in the basis Λ of S (ST), in [DLP] we manage the gaps in the indices of the looping generators of elements inΣ, obtaining elements in the augmented H n ( q )-module Λ aug , denoted by Λ aug | H n . We needto emphasize on the fact that the “managing the gaps” procedure, allows the performance ofbbm’s to take place on any moving strand. Then, these equations are shown to be equivalentto equations obtained from elements in the H n ( q )-module Λ, denoted by Λ | H n , by performingbbm’s on any moving strand. We finally eliminate the braiding “tails” from elements in Λ | H n and reduce the computations to the set Λ, where the bbm’s are performed on any movingstrand (see [DLP]). Thus, in order to compute S ( L ( p, aug and show that the system of equations btained from elements in Λ by performing bbm’s on any moving strand, is equivalent to thesystem of equations obtained by performing bbm’s on the first moving strand of elements inΛ aug . It is worth mentioning that although Λ aug ⊃ Λ, the advantage of considering elements inthe augmented set Λ aug is that we restrict the performance of the braid band moves only on thefirst moving strand and, thus, we obtain less equations and more control on the infinite system(1).The above are summarized in the following sequence of equations: S ( L ( p, S (ST) , a ∈ B ,n , ∀ i = S (ST) , s ′ ∈ Σ ′ n == S (ST) , s ∈ Σ n = S (ST) <λ ′ − bbm i ( λ ′ ) > , λ ′ ∈ Λ aug | H n , ∀ i == S (ST) <λ ′′ − bbm i ( λ ′′ ) > , λ ′′ ∈ Λ | H n , ∀ i = S (ST) <λ − bbm i ( λ ) > , λ ∈ Λ , ∀ i == S (ST) <µ − bbm ( µ ) > , µ ∈ Λ aug . Namely, we have:
Theorem 8. ([DLP, DL4])i . S ( L ( p, S (ST) <λ − bbm i ( λ ) > , λ ∈ Λ , ∀ i. ii . S ( L ( p, S (ST) <µ − bbm ( µ ) > , µ ∈ Λ aug . Relating the “positive” and “negative” sub-modules of S ( L ( p, B + to the infinite system obtained by performing bbm’s on elements in B − . Wepresent now the sets B + and B − in open braid form:(9) Λ + := { t k t k . . . t k n n , k i ∈ N \{ } : k i ≤ k i − , ∀ i } Λ − := { t k t k . . . t k n n , k i ∈ Z \ N : k i ≤ k i − , ∀ i } We now augment the sets Λ + , Λ − by allowing arbitrary exponents in monomials in the t i ’s,and we define the the corresponding subsets of Λ + , Λ − of level k as follows: Definition 5.
We define the “positive” subset of Λ aug ( k ) :(10) Λ aug + ( k ) := { t k t k . . . t k n n , k i ∈ N \{ }} . and the the “negative” subset of Λ aug ( k ) :(11) Λ aug − ( k ) := { t k t k . . . t k n n , k i ∈ Z \ N } . The infinite system of equations obtained by performing ± - bbm ’s on elements in Λ aug + isrelated to the infinite system of equations obtained by performing ∓ - bbm ’s on elements in Λ aug − by the following maps: Definition 6. (i) We define the automorphism f : Λ aug → Λ aug such that: ( τ · τ ) = f ( τ ) · f ( τ ) , ∀ τ , τ ∈ Λ aug t ki t − ki , ∀ i ∈ N ∗ , ∀ k ∈ Z \ N σ i σ − i , ∀ i ∈ N ∗ (ii) We define the map I : R [ z ± , s k ] → R [ z ± , s k ], k ∈ Z such that: I ( τ + τ ) = I ( τ ) + I ( τ ) , ∀ τ , τ I ( τ · τ ) = I ( τ ) · I ( τ ) , ∀ τ , τ s − k s k , ∀ k ∈ N s p − k s p + k , ∀ k : 0 ≤ k ≤ pz λ · zq ± q ∓ λ k z λ k +1 z , ∀ k We now state the main result of this paper:
Theorem 9.
The equations obtained by imposing on the invariant X relations coming from theperformance of a ± - bbm on an element τ in Λ aug + are equivalent to the image of the equationsobtained by performing ∓ - bbm on its corresponding element f ( τ ) in Λ aug − under I . That is: I (cid:16) X d f ( τ ) = X \ bbm ∓ ( f ( τ )) (cid:17) ⇔ X b τ = X \ bbm ± ( τ ) Equivalently we have:
Corollary 1.
The following diagram commutes: Λ aug ( k ) ∋ τ bbm ± → bbm ± ( τ ) ⇒ X b τ = X \ bbm ( τ ) , X b τ = X \ bbm − ( τ ) l f ↑ I ↑ I Λ aug ( − k ) ∋ f ( τ ) bbm ∓ → bbm ∓ ( τ ) ⇒ X d f ( τ ) = X \ bbm − ( f ( τ )) , X d f ( τ ) = X \ bbm +1 ( f ( τ )) Definition 7.
We will say that the elements bbm ± τ and bbm ∓ f ( τ ) are “symmetric” with respectto the sign of their exponents (or just “symmetric”), although the loop generator t p appears inboth. Moreover, an element in H n ( q ) will be called braiding “tail” and two braiding “tails” willbe called “symmetric” with respect to the sign of their exponents, if one is obtained from theother by changing σ ± i to σ ∓ i .3.1. The infinite system.
Let τ k ,m ,m ∈ Λ aug ( k ) , that is P mi =0 k i = k . We present some results onthe infinite system of equations (1): X \ τ k ,m ,m = X \ t p τ k ,m ,m +1 σ X \ τ k ,m ,m = X \ t p τ k ,m ,m +1 σ − which is equivalent to (see Eq. (13) and (14)): igure 6. t − t ′ = tt ′ − . tr ( τ k ,m ,m ) = z · λ P mj =0 ( j +1) k j · tr ( t p τ k ,m ,m +1 g ) tr ( τ k ,m ,m ) = z · λ P mj =0 ( j +1) k j − · tr ( t p τ k ,m ,m +1 g − )We have the following: Proposition 1.
The unknowns s , s , . . . of the system commute.Proof. Consider the set of all permutations of the set S = k , . . . k n and let ϕ be a bijection fromthe set S to itself. We consider now the elements α = t ′ i k . . . t ′ i n k n and β = t ′ i ϕ ( k ) . . . t ′ i n ϕ ( k n ) ,where 0 ≤ i ≤ i ≤ . . . ≤ i n of the basis of S ( ST ). We have that: tr ( α ) = s k n . . . s k and tr ( β ) = s ϕ ( k n ) . . . s ϕ ( k ) . We compute the invariant X on the closures b α, b β of α and β ,respectively, and we obtain: X ( b α ) = [ − − λq √ λ ] n − √ λ tr ( α ) = [ − − λq √ λ ] n − s k n . . . s k and X ( b β ) =[ − − λq √ λ ] n − √ λ tr ( β ) = [ − − λq √ λ ] n − s ϕ ( k n ) . . . s ϕ ( k ) . Now, the n -component link b α is isotopic to b β in ST , as illustrated in Figure 6 for the case of two components. So, we have that X ( b α ) = X ( b β ) ,equivalently,(12) s k n . . . s k = s ϕ ( k n ) . . . s ϕ ( k ) and so the unknowns of the system commute.Equation 12 holds for any subset S of Z and for any permutation φ of S , hence the unknowns s i of the system (1) must commute. (cid:3) The modules Λ aug + < bbm > and Λ aug − < bbm > . From now on, we will consider all braid band movesto take place on the first moving strand of elements in Λ aug and we will denote by bbm + ( τ ) theresult of the performance of a positive bbm and bbm − ( τ ) will correspond to the result of theperformance of a negative bbm on τ ∈ Λ aug .Let τ := τ k ,n ,n ∈ Λ aug + and perform a bbm + . We have that bbm + ( τ ) = t p τ k ,n ,n +1 σ and we obtain the equation: X b τ = X \ bbm + ( τ ) ⇔ tr ( τ k ,n ,n ) = − − λq √ λ (1 − q ) √ λ n P i =0 i +1) k i · tr ( t p τ k ,n ,n +1 g )and since λ = z +1 − qqz , we obtain tr ( τ k ,n ,n ) = λ n P i =0 ( i +1) k i z · tr ( t p τ k ,n ,n +1 g )Consider now f ( τ ) := τ − k ,n ,n ∈ Λ aug − and perform a bbm − . We have that bbm − ( f ( τ )) = t p τ − k ,n ,n +1 σ − and we obtain the equation: X d f ( τ ) = X \ bbm − ( f ( τ )) ⇔ tr ( τ − k ,n ,n ) = − − λq √ λ (1 − q ) √ λ − n P i =0 i +1) k i · tr ( t p τ k ,n ,n +1 g ), that is:(14) tr ( τ − k ,n ,n ) = λ − − n P i =0 ( i +1) k i z · tr ( t p τ − k ,n ,n +1 g − )In order to prove Corollary 1, we first prove that the image of the coefficient in Equation 14under the map I is equal to the coefficient of Equation 13. Indeed we have the following: Lemma 1. I λ − − n P i =0 ( i +1) k i z = λ n P i =0 ( i +1) k i z Proof.
We have that: I λ − − n P i =0 ( i +1) k i z = λ n P i =0 ( i +1) k i − z = λ n P i =0 ( i +1) k i z (cid:3) Remark 3.
Lemma 1 demonstrates the motivation for the Definition 6(ii) on λ k z , where λ k z I λ k +1 z .We now relate tr ( τ ) to I ( tr ( f ( τ ))) using the fact that relations used in order to convertelements in Λ aug + to sums of elements in (Λ ′ ) aug + , where (Λ ′ ) aug + is defined as monomials in t ′ i ’s with positive exponents, are “symmetric”, as shown for example below: σ i = q σ − i + ( q − −
1) & σ i = ( q − σ i + qσ − i = q − σ i + ( q −
1) & σ − i = ( q − − σ − i + q − Moreover, these relations lead to q I q − in Definition 6(ii), and the rule z I z · λ comes fromthe fact that tr ( σ i ) = z , while tr ( σ − i ) = q − z + ( q − −
1) = z +1 − qq = λ · z . The readeris now referred to [La1, La2, DL2, DL3, DL4, DLP] for other “symmetric” relations, and alsofor more details on the techniques applied in order to obtain the infinite change of basis matrixrelating the sets Λ and Λ ′ .We are now in position to translate Theorems 5, 6 and 7 in the context of this paper: Theorem 5 suggests that an element τ in Λ aug + can be written as a sum of its homologousword τ ′ in (Λ ′ ) aug + , the element τ followed by a braiding “tail” and monomials in Σ whichare of lower order than the initial monomial τ .Similarly, an element T in Λ aug − can be written as a sum of its homologous word T ′ in (Λ ′ ) aug − , the element T followed by a braiding “tail” and monomials in Σ which areof lower order than the initial monomial T .Moreover, as explained above, corresponding coefficients in these two processes willbe “symmetric” and the braiding “tails” in τ , after Theorem 5 is applied, will be “sym-metric” to the braiding “tails” in T . • For the elements τ and T that are followed by braiding “tails” in H n ( q ), we applyTheorem 6 and we have that corresponding coefficients will be “symmetric” only for theterms involving the parameter q , while for z , the reader is referred to the discussion afterRemark 3. • By Theorem 7 the order of the exponents in an element in Λ aug can be altered, leadingto elements of lower (or even greater) order. For the braiding “tail” occurring afterapplying Theorem 7, we apply Theorem 6 again, and this procedure will eventually stopand the result will be a sum of elements in Λ aug of lower order than the initial element.For more details of how this procedure terminates the reader is referred to [DL2] and[DL3].We now have the following result:
Proposition 2.
For τ ∈ Λ aug + ( k ) , where k ∈ N , the following relation holds: tr ( τ ) = I ( tr ( f ( τ ))) , where f ( τ ) ∈ Λ aug − ( − k ) .Proof. Let τ := τ k ,n ,n ∈ Λ aug + ( k ) , k ∈ N and f ( τ ) := τ − k ,n ,n ∈ Λ aug − ( − k ) . In order to evaluate tr ( τ ) weuse the inverse of the change of basis matrix and express τ as a sum of elements in (Λ ′ ) aug + ( k ) ,i.e. τ b ∼ = P i A i τ ′ i , where A i coefficients in C [ q ± , z ± ] for all i , such that ∃ j : τ ′ j = τ ′ ∼ τ and τ ′ i < τ ′ , for all i = j . Following the same steps in order to express f ( τ ) as a sum of elements in(Λ ′ ) aug − ( − k ) , we prove that we obtain that f ( τ ) b ∼ = P i B i f ( τ ′ i ), where B i coefficients in C [ q ± , z ± ]for all i , such that ∃ j : f ( τ ′ j ) = f ( τ ′ ) ∼ f ( τ ) and f ( τ ′ i ) < f ( τ ′ ), for all i = j , and also that f ( B i ) = A i , for all i . We prove that by strong induction on the order of τ .The base of induction is t k ∈ Λ aug + ( k ) , where tr ( t k ) = s k and f ( t k ) = t − k and tr ( t − k ) = s − k .We observe that s − k I s k by Definition 6, and thus I (cid:0) tr (cid:0) f ( t − k ) (cid:1)(cid:1) = tr (cid:0) t k (cid:1) .Assume now that I ( tr ( f ( τ i ))) = tr ( τ i ), for all τ i < τ ∈ Λ aug + ( k ) . Then, for τ we have that: := τ k ,n ,n := t k t k . . . t k n n = t k . . . t k n − n · ( σ n . . . σ t σ . . . σ n ) == q τ k ,n − ,n − · t k n − n · ( σ n . . . σ t σ − σ . . . σ n ) | {z } A ++ ( q − t k +1 · τ k ,n − ,n − · t k n − n · ( σ n . . . σ σ . . . σ n ) | {z } B and for f ( τ ) we obtain: f ( τ ) := τ − k ,n ,n := t − k t − k . . . t − k n n = t − k . . . t − k n +1 n · ( σ − n . . . σ − t − σ − . . . σ − n ) == q − τ − k ,n − ,n − · t − k n +1 n · ( σ − n . . . σ − σ t σ − . . . σ − n ) | {z } C ++ ( q − − t − k +1 · τ − k ,n − ,n − · t − k n +1 n · ( σ − n . . . σ − . . . σ − n ) | {z } D Observe now that D = f ( B ) and according to Definition 2 we have that t k +1 · τ k ,n − ,n − · t k n − n · ( σ n . . . σ σ . . . σ n ) < τ. Thus, from the induction step we obtain that I ( tr ( D )) = tr ( B ).We now apply Theorems 7, 8, 9 & 10 in [DL2] (relations used for the change of basis matrixand which are Theorems 5, 6, 7 & Eq.(8) in this paper) on A and C and we use the inductionstep to reduce the complexity of the relations obtained each time a lower order term appearsin the relations. As explained in [DL2, DL4], we will eventually obtain the homologous word in(Λ ′ ) aug + k for τ (coming from A ) and the homologous word in (Λ ′ ) aug − − k for f ( τ ) (coming from C ),with “symmetric” coefficients. Hence, tr ( τ ) = I ( tr ( f ( τ ))) . (cid:3) In order to proceed with the proof of Corolarry 1, we will need the following relations(Lemma 2 [La2]):(15) t kn σ n = ( q − k − P j =0 q j t jn − t k − jn + q k σ n t kn − , k ∈ N t − kn σ − n = ( q − − − k +1 P j =0 q j t jn − t − k − jn + q − k σ n t − kn − σ − n , k ∈ N and the following lemmas: emma 2. For n, k ∈ N , the following relations hold: i . t kn = k − P j =1 q j − ( q − t jn − t k − jn · σ n + q k − σ n t kn − σ n ii . t − kn = − k +1 P j = − q j +1 ( q − − t jn − t − k − jn · σ − n + q − k +1 σ − n t − kn − σ − n Proof.
We only prove relations (i). Relations (ii) follow similarly. For k = 2 we have that: t n = σ n t n − σ n t n − σ n = ( q − σ n t n − σ n t n − σ n + q σ n t n − σ n == ( q − t n − t n σ n + q σ n t n − σ n For k ∈ N we have: t kn = t k − n · t n = ( q − t k − n t n − σ n + q t k − n σ n t n − σ n = Eq.
15= ( q − t k − n t n − σ n + q ( q − k − P j =0 q j t jn − t k − − jn t n − σ n ++ q k − σ n t k − n − t n − σ n == k − P j =1 q j − ( q − t jn − t k − jn · σ n + q k − σ n t kn − σ n (cid:3) Lemma 3.
For n, k ∈ N , the following relations hold: i . t kn σ n +1 = q − k +1 σ − n +1 t kn +1 + k − P j =1 q − j +1 ( q − − t k − jn t jn +1 ii . t − kn σ − n +1 = q k − σ n +1 t − kn +1 + k − P j =1 q j − ( q − t − k + jn t − jn +1 Proof.
We prove relations (i) by induction on k ∈ N . Relations (ii) follow similarly.For k = 1 we have that t n σ n +1 = σ − n +1 t n +1 , which is true. Assume now that the relationholds for k −
1. Then, for k we have: t kn σ n +1 = t n t k − n σ n +1 ind. = step = q − k +2 t n σ − n +1 t k − n +1 + k − P j =1 q − j +1 ( q − − t k − jn t jn +1 == q − k +1 t n σ n +1 t k − n +1 + q − k +2 ( q − − t n t k − n +1 + k − P j =1 q − j +1 ( q − − t k − jn t jn +1 == q − k +1 σ − n +1 t kn +1 + k − P j =1 q − j +1 ( q − − t k − jn t jn +116 We are now in position to prove the following lemma that serves as the basis of the induc-tion applied in the final result of this section, Proposition 3, which will conclude the proof ofCorollary 1.
Lemma 4.
Let t k bbm ± −→ t p t k σ ± and f ( t k ) := t − k bbm ∓ −→ t p t − k σ ∓ . Then, the followingrelations hold: I (cid:16) tr ( t p t − k σ ∓ ) (cid:17) = tr ( t p t k σ ± ) . Proof.
We only prove that I (cid:0) tr ( bbm − ( f ( t k ))) (cid:1) = tr ( bbm + ( t k )), by strong induction on theorder of bbm − ( f ( t k )). The case I (cid:0) tr ( bbm + ( f ( t k ))) (cid:1) = tr ( bbm − ( t k )) follows similarly.The base of induction is the case k = 1, i.e. I (cid:0) tr ( t p t − σ − ) (cid:1) = tr ( t p t σ ). We have that: tr ( t p t σ ) = ( q − tr ( t p t ) + q tr ( t p σ t ) == q ( q − tr ( t p t ′ ) + ( q − tr ( t p +1 σ ) + q tr ( t p +1 σ ) == q ( q − s s p + ( q − z s p +1 + qz s p +1 and tr ( t p t − σ − ) = ( q − − tr ( t p t − ) + q − tr ( t p σ − t − ) == q − ( q − − tr ( t p t ′ − ) + ( q − − tr ( t p − σ − ) ++ q − tr ( t p − σ − ) == q − ( q − − s − s p + q − ( q − − z s p − ++ ( q − − s p − + q − ( q − − s p − + q − z s p − Moreover: I (cid:16) tr ( t p t − σ − ) (cid:17) = I (cid:0) q − ( q − − s − s p (cid:1) + I (cid:0) q − ( q − − z s p − (cid:1) ++ I (cid:0) ( q − − s p − (cid:1) + I (cid:0) q − ( q − − s p − (cid:1) ++ I (cid:0) q − z s p − (cid:1) == q ( q − s s p + q ( q − λz s p +1 ++ ( q − s p +1 + q zλ s p +1 ++ q ( q − ) s p +1 ⇒ (cid:0) tr ( t p t − σ − ) (cid:1) = q ( q − s s p ++ h q ( q − z +1 − qqz + ( q − + q z z +1 − qqz + q ( q − i s p +1 == q ( q − s s p + ( q − z s p +1 + qz s p +1 == tr ( t p t k σ )Assume now that I (cid:16) tr ( t p − i t − k − i σ − ) (cid:17) = tr ( t p + i t k − i σ ), for all 0 < i < k . Then, for i = 0we have: tr (cid:0) t p t k σ (cid:1) Eq.
15= ( q − k − P j =0 q j t p + j t k − j + q k z tr ( t p + k ) == ( q − k − P j =0 q j tr ( t p + j t k − j ) + q k z s p + k tr (cid:16) t p t − k σ − (cid:17) Eq.
15= ( q − − − k +1 P j =0 q j tr ( t p + j t − k − j ) ++ q − k − z tr ( t p − k ) + q − k ( q − − tr ( t p − k ) == ( q − − − k +1 P j =0 q j tr ( t p + j t − k − j ) + (cid:0) q − k − z + q − k ( q − − (cid:1) s p − k We have that I (cid:0)(cid:0) q − k − z + q − k ( q − − (cid:1) s p − k (cid:1) = (cid:0) q k +1 λ z + q k ( q − (cid:1) s p + k == (cid:16) q k +1 z +1 − qqz z + q k ( q − (cid:17) s p + k = q k z s p + k Moreover, the terms t p + j t k − j are of lower order than t p t k for all j = 0 and thus, from theinduction step we have that: I (cid:16) tr ( t p + j t k − j ) (cid:17) = tr ( t p + j t − k − j ) , for all j = 0 . For j = 0 we have: p t k = t p t k − t = ( q − t p +1 t k − σ + q t p t k − σ t σ == ( q − t p +1 t k − σ + q ( q − k − P j =0 q j t p + j +2 t k − − j σ + q k − t p σ t k σ == ( q − t p +1 t k − σ | {z } A + q ( q − k − X j =0 q j t p + j +2 t k − − j σ | {z } B ++ q k − t p t ′ k | {z } C + q k − ( q − t p σ t k | {z } D and t p t − k = t p t − k +21 t − = ( q − − t p − t − k +11 σ − + q − t p t − k +21 σ − t − σ − == ( q − − t p − t − k +11 σ − + q ( q − − k +3 P j =0 q j t p + j − t − k +2 − j σ − ++ q − k +2 t p σ − t − k σ − == ( q − − t p − t − k +11 σ − | {z } A ′ + q ( q − − k +3 X j =0 q j t p + j − t − k +2 − j σ − | {z } B ′ ++ q − k +1 t p t ′ − k | {z } C ′ + q − k +2 ( q − − t p t − k σ − | {z } D ′ Observe now now that in A the term t p +1 t k − σ is of lower order than t p t k and also that theterms A, A ′ are “symmetric”. From the induction step we have that I ( tr ( A ′ )) = tr ( A ). Forthe same reasons I ( tr ( B ′ )) = tr ( B ). Finally we have that: tr ( C ) = q k − tr ( t p t ′ k ) = q k − s p s k ⇒ I (cid:0) q − k +1 s p s − k (cid:1) = q k − s p s k tr ( C ′ ) = q − k +1 t p t ′ − k = q − k +1 s p s − k and that tr ( D ) = q k − ( q − tr ( t p + k σ ) = q k − ( q − z s p + k tr ( D ′ ) = q − k +2 ( q − − tr ( t p − k σ − ) = q − k +1 ( q − − z s p − k + q − k +2 ( q − − s p − k ( tr ( D ′ )) = (cid:0) q k − ( q − λ z + q k − ( q − (cid:1) s p − k = (cid:16) q k − ( q − z +1 − qqz z + q k − ( q − (cid:17) s p − k = (cid:0) q k − ( q −
1) ( z + 1 − q ) + q k − ( q − (cid:1) s p − k = q k − ( q − z s p + k ⇒ I ( tr ( D ′ )) = tr ( D )The proof is now concluded. (cid:3) We are now ready to state and prove the final proposition that concludes the proof of Corol-lary 1.
Proposition 3.
Let τ ∈ Λ aug + ( k ) , where k ∈ N , and bbm + ( τ ) the result of the performance ofa positive braid band move on τ . Let also f ( τ ) ∈ Λ aug − ( − k ) and bbm − ( f ( τ )) the result of theperformance of a negative braid band move on f ( τ ) . Then, the following relation holds: I ( tr ( bbm ∓ ( f ( τ )))) = tr ( bbm ± ( τ )) , where τ ∈ Λ aug + ( k ) . Proof.
We only prove that I ( tr ( bbm − ( f ( τ )))) = tr ( bbm + ( τ )), by strong induction on theorder of bbm − ( f ( τ )). The case I ( tr ( bbm + ( f ( τ )))) = tr ( bbm − ( τ )) follows similarly. Let τ = t k t k . . . t k n n , f ( τ ) = t − k t − k . . . t − k n n ,bbm + ( τ ) = t p t k . . . t k n − n t k n n +1 σ , bbm − ( f ( τ )) = t p t − k . . . t − k n − n t − k n n +1 σ − . The base of induction is Lemma 4. Assume now that I ( tr ( bbm − ( f ( τ i )))) = tr ( bbm + ( τ i )),for all τ i < τ . Then, for τ we have that: tr ( t p t k . . . t k n − n t k n n +1 σ ) = tr ( t p ( t k · σ ) . . . t k n − n t k n n +1 ) Eq. q − k − X j =0 q j tr (cid:16) t p + j t k − j τ k ,n ,n +1 (cid:17)| {z } A + q k tr (cid:16) t p + k τ k ,n ,n +1 · σ (cid:17)| {z } B , and r ( t p t − k . . . t − k n − n t − k n n +1 σ − ) = tr ( t p ( t − k · σ − ) . . . t − k n − n t − k n n +1 ) Eq. q − − − k +1 X j =0 q j tr (cid:16) t p + j t − k − j τ − k ,n ,n +1 (cid:17)| {z } A ′ ++ q − k tr (cid:16) t p − k τ − k ,n ,n +1 · σ − (cid:17)| {z } B ′ Observe now that the terms B and B ′ are “symmetric” and also that the term t p + k τ k ,n ,n +1 · σ in B has a “gap” in the indices, and thus, it is of lower order than τ according to Defini-tion 2(b)(ii)( α ). Note also that in A , the terms t p + j t k − j τ k ,n ,n +1 are “symmetric” with thecorresponding terms in A ′ for all j , and that for j = 0, all terms are of lower order than τ . Thus,from the induction hypothesis, we have that I ( tr ( B ′ )) = tr ( B ) I ( tr ( A ′ )) = tr ( A ) for all j = 0 . For j = 0 in A we obtain the element tr (cid:16) t p τ k ,n ,n +1 (cid:17) and for j = 0 in A ′ we obtain tr (cid:16) t p τ − k ,n ,n +1 (cid:17) .These element’s are “symmetric” and we have the following: tr (cid:16) t p τ k ,n ,n +1 (cid:17) = tr (cid:16) t p τ k ,n − ,n t k n n +1 (cid:17) = k n − X j =1 q j − ( q − tr (cid:16) t p t k ,n − ,n t jn t k n − jn +1 · σ n +1 (cid:17)| {z } A + q k n t p τ k ,n − ,n σ n +1 t k n n σ n +1 | {z } B and tr (cid:16) t p τ − k ,n ,n +1 (cid:17) = tr (cid:16) t p τ − k ,n − ,n t − k n n +1 (cid:17) = k n − X j =1 q j +1 ( q − − tr (cid:16) t p t − k ,n − ,n t jn t − k n + jn +1 · σ − n +1 (cid:17)| {z } A ′ = q − k n t p τ − k ,n − ,n σ − n +1 t − k n n σ − n +1 | {z } B ′ We observe again that the terms A and A ′ are “symmetric” and of lower order than τ, f ( τ )and thus, from the induction hypothesis we have that I ( A ′ ) = tr ( A ). For the terms B, B ′ weonly have that the coefficients are “symmetric”. The idea is to change the order of particularexponents that will lead to lower order terms in the resulting sum, when applying Theorem 7.We demonstrate this technique for the case k n − < k n : = q k n tr (cid:16) t p τ k ,n − ,n − t k n − n σ n +1 t k n n σ n +1 (cid:17) Lem. k n − − P j =1 tr ( ... ) + q k n − k n − +1 tr (cid:16) t p τ k ,n − ,n − σ − n +1 t k n − n +1 t k n n σ n +1 (cid:17) == k n − − P j =1 tr ( ... ) + q k n − k n − +1 tr (cid:16) t p τ k ,n − ,n − t k n − n +1 t k n n (cid:17) Similarly for B ′ we obtain: B ′ Lem. k n − − X j =1 tr ( ... ) + q − k n + k n − − tr (cid:16) t p τ − k ,n − ,n − t − k n − n +1 t − k n n (cid:17) Monomials in t i ’s in the sums in these relations are “symmetric” and of lower order thanthe initial monomial, and, assuming k n − < k n , same is true for t p τ k ,n − ,n − t k n − n +1 t k n n . The resultfollows from the induction step. (cid:3) Lemma 1 and Propositions 2 and 3 are summarized in the following diagram: tr (cid:16) f ( τ k ,n ,n ) (cid:17) bbm ∓ = λ − − n P i =0 ( i +1) ki z · tr ( t p τ − k ,n ,n +1 g − ) f ↑ P rop. I ↑ Lem. I ↑ P rop. tr ( τ k ,n ,n ) bbm ± = λ n P i =0 ( i +1) ki z · tr ( t p τ k ,n ,n +1 g )The proof of Corollary 1, and thus, the proof of the main theorem, Theorem 9, is nowconcluded. 4. Toward the HOMFLYPT skein module of L ( p, S ( L ( p, aug + and we prove that the infinite system of equations splits into self-contained subsystems. Moreprecisely: Lemma 5.
Let τ ∈ Λ aug + k . Then tr ( τ ) = P i f i ( q, z ) s u ,v ,v , where s u ,v ,v := s u s u . . . s u v v , suchthat u i ∈ N for all i and v P i =1 i · u i = k .Proof. It derives directly from the change of basis matrix and the fourth rule of the trace. (cid:3)
Corollary 2.
For k ∈ N we obtain an infinite self-contained system of equations by performingbbm’s on elements in Λ aug + ( k ) . That is, the system (1) splits into infinitely many self-containedsubsystems of equations. emark 4. Note that if instead of considering elements in Λ aug + ( k ) and performing bbm ’s, weconsidered elements in the set Λ aug ( k ) , k ∈ Z , and perform bbm ’s, then the infinitely many sub-systems of equations would again be self-contained, but they would be of infinite dimension, i.e.the number of equations and unknowns in each sub-system would be infinite. This is due to thefact that the exponents in the monomials in t i ’s in Λ aug ( k ) are arbitrary (positive and negative). Wecall such monomials, monomials of “mixed” exponents and we deal with the equations obtainedby performing bbm’s on these elements in a sequel paper. For more details the reader is referredto [D].4.1. Potential bases for the submodules Λ aug +
The set (16) n t ′ k t ′ k . . . t ′ nk n , where n, k i ∈ N : 0 ≤ k i ≤ p − o is a generating set of the module Λ aug +
From Theorem 8 we have that Λ aug +
The set n t ′ k t ′ k . . . t ′ nk n , where n, k i ∈ N : 0 ≤ k i ≤ p − o forms a basis for Λ aug +
The set n t ′ k t ′ k . . . t ′ nk n , where n, k i ∈ Z \ N : 0 ≥ k i ≥ − p + 1 o is a generating set of the module Λ aug −
Conjecture 2.
The set n t ′ k t ′ k . . . t ′ nk n , where n, k i ∈ N : 0 ≥ k i ≥ − p + 1 o forms a basis for Λ aug −
We now consider the subsystems of equations obtained by performingbraid band moves on elements in Λ aug − ( k< . We use Theorem 9 and the equations obtained fromelements in Λ aug + , presented above. • For k = − t − ∈ Λ aug − ( − and: t − bbm ± → t p t − σ ± ⇔ s p − = s − & s p s − = a ′ s . For the elements in Λ aug − ( − we have: t − bbm → t p t − σ ± t − t − bbm → t p t − t − σ ± and: s p − = A ′ s − + A ′ s − s − s p = B ′ s − + B ′ s − s − s p − = C ′ s − + C ′ s − s p s − = D ′ s − + D ′ s − where A ′ i , B ′ i , C ′ i , D ′ i ∈ C , ∀ i .Observe now for example that the unknown s p − is equal to s − and that the unknown s p − can be written as a combination of s − and s − . Thus, s p − , s p − will not be in the basis of S ( L ( p, aug + ( k> , and results from [D], suggest that the followingset forms a basis for S ( L ( p, n t ′ k t ′ k . . . t ′ nk n , where n, k i ∈ Z : − p/ ≤ k i < p/ o It is worth mentioning that the same set was obtained and proved to be a basis for S ( L ( p, L ( p, q ) , q > References [D]
I. Diamantis , The HOMFLYPT skein module of the lens spaces L ( p,
1) via braids,
PhD thesis , NationalTechnical University of Athens, 2015.[D2]
I. Diamantis , (2019) An Alternative Basis for the Kauffman Bracket Skein Module of the Solid Torusvia Braids. In: Adams C. et al. (eds) Knots, Low-Dimensional Topology and Applications. KNOTS162016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham.[D3]
I. Diamantis , The Kauffman bracket skein module of the handlebody of genus 2 via braids,
J. KnotTheory and Ramifications , , No. 13, 1940020 (2019).[DL1] I. Diamantis, S. Lambropoulou , Braid equivalences in 3-manifolds with rational surgery description,
Topology and its Applications , (2015), 269-295.[DL2] I. Diamantis, S. Lambropoulou , A new basis for the HOMFLYPT skein module of the solid torus,
J.Pure Appl. Algebra
Vol. 2 (2016), 577-605.[DL3]
I. Diamantis, S. Lambropoulou , The braid approach to the HOMFLYPT skein module of the lensspaces L ( p, Algebraic Modeling ofTopological and Computational Structures and Application , (2017).[DL4]
I. Diamantis, S. Lambropoulou , An important step for the computation of the HOMFLYPT skeinmodule of the lens spaces L ( p,
1) via braids,
J. Knot Theory and Ramifications , , No. 11, 1940007(2019).[DL5] I. Diamantis, S. Lambropoulou , The HOMFLYPT skein module of the lens spaces L ( p,
1) via braids,in preparation.[DLP]
I. Diamantis, S. Lambropoulou, J. H. Przytycki , Topological steps on the HOMFLYPT skeinmodule of the lens spaces L ( p,
1) via braids,
J. Knot Theory and Ramifications , , No. 14, (2016).[FYHLMO] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu , A newpolynomial invariant of knots and links,
Bull. Amer. Math. Soc. , (1985), 239-249.[GM] B. Gabrovˇsek, M. Mroczkowski , The Homlypt skein module of the lens spaces L ( p, Topology andits Applications , (2014), 72-80. HK]
J. Hoste, M. Kidwell , Dichromatic link invariants,
Trans. Amer. Math. Soc. (1990), No. 1,197-229.[HP]
J.Hoste, J.H.Przytycki , A survey of skein modules of 3-manifolds.
Knots 90 (Osaka, 1990) , deGruyter, Berlin, (1992) 363?-379.[Jo]
V. F. R. Jones , A polynomial invariant for links via Neumann algebras,
Bull. Amer. Math. Soc.
S. Lambropoulou , Knot theory related to generalized and cyclotomic Hecke algebras of type B , J.Knot Theory and its Ramifications , No. 5, (1999) 621-658.[La2] S. Lambropoulou , Solid torus links and Hecke algebras of B-type,
Quantum Topology ; D.N. Yetter Ed.;World Scientific Press, (1994), 225-245.[LR1]
S. Lambropoulou, C.P. Rourke (2006), Markov’s theorem in 3-manifolds,
Topology and its Applica-tions , (1997) 95-122.[LR2] S. Lambropoulou, C. P. Rourke , Algebraic Markov equivalence for links in 3-manifolds,
CompositioMath. (2006) 1039-1062.[P]
J. Przytycki , Skein modules of 3-manifolds,
Bull. Pol. Acad. Sci.: Math. ,
39, 1-2 (1991), 91-100.[PT]
J. H. Przytycki, P. Traczyk , Invariants of links of Conway type,
Kobe J. Math. (1987), 115-139.[Tu] V.G. Turaev , The Conway and Kauffman modules of the solid torus,
Zap. Nauchn. Sem. Lomi (1988), 79–89. English translation:
J. Soviet Math. (1990), 2799-2805.
International College Beijing, China Agricultural University, No.17 Qinghua East Road, Haid-ian District, Beijing, 100083, P. R. China.
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