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The Colored Jones Polynomial and Aj Conjecture

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The Colored Jones Polynomial and Aj Conjecture THE COLORED JONES POLYNOMIAL AND AJ CONJECTURE THANG T. Q. LE^ Abstract. We present the basics of the colored Jones polynomials and discuss the AJ conjecture. Supported in part by National Science Foundation. 1 2 THANG T. Q. LE^ 1. Jones polynomial 1.1. Knots and links in R3 ⊂ S3. Fix the standard R3. An oriented link L is a compact 1-dimensional oriented smooth submanifold of R3 ⊂ S3. Denote by #L the number of connected components of L. A link of 1 component is called a knot. By convention, the empty set is also considered a link. A framed oriented link L is a link equipped with a smooth normal vector field V , which 3 is a function V : L ! R , such that V (x) is not in the tangent space TxL for every x 2 L. 3 3 One should consider V (x) as an element in the tangent space TxR of R at x. 3 3 Two oriented links L1 and L2 are equivalent if there is a smooth isotopy h : R ! R such 3 3 that h(L1) = (L2). Here h is a smooth isotopy if there is smooth map H : R × [0; 1] ! R 3 3 such that for every t 2 [0; 1], ht(x): R ! R defined by ht(x) = H(x; t), is a diffeomorphism 3 of R , and h0 = id; h1 = h. Similarly, two framed oriented links (L1;V1) and (L2;V2) are equivalent if there is a smooth 3 3 isotopy h : R ! R such that h(L1) = (L2) and for every x 2 L1, dhx(V1(x)) = V2(h(x)): Here dhx is the derivative of h at x. A (framed) link is ordered if there is an order on the set of its components. Then the equivalence relation is required to preserve the order. The framing of a component can be specified by thickening the component to a ribbon in the direction of the framing vector. Usually we don't distinguish between a link and its equivalence class. Un-oriented links, un-oriented framed links and their equivalence classes are defined sim- ilarly. A link invariant is a map I : fequivalence classes of linksg ! R; where R is a set. 3 Example 1.1. For unoriented unframed links, the link group π1(R n L) is a link invariant. 1.2. Link diagram, blackboard framing. One often studies an (oriented or unoriented) link L by its diagram on R2, which is the projection D of L onto R2 (in general position), to- gether with the \over/under" information at each crossing point. An (oriented) link diagram defines an equivalence class of (oriented) links. A link diagram comes with the blackboard framing, in which the framing vectors are in the plane R2. If the framed link determined by a link diagram D with the blackboard framing is isotopic the a framed link L, we say that D is a blackboard diagram of L. It is known that two unoriented unframed link diagrams define the same class of links if and only if they are related by a sequence of Reidemeister moves RI, RII, and RIII and isotopies of the plane. The Reidemeister moves are listed in Figure 1 and 2. For framed unoriented link diagrams one replaces RI by RIf . For oriented links one allows all possible orientations of the strands in the figures. For details, see e.g [BZ, Oh]. Thus, the map associating an unoriented unframed link diagram to its link class descends to an isomorphism of sets fequiv. classes of linksg ! flink diagramsg=(RI,RII,RIII, isotopy of R2): COLORED JONES POLYNOMIAL 3 Figure 1. Reidemeister move RI on the left and RIf on the right. Figure 2. Reidemeister move RII on the left and RIII on the right. The mirror image of link diagram D is the result of switching all the crossings of D from over to under and vice versa. 1.3. Sign of a crossing, linking number, writhe. Up to isotopy there are two types of crossings of oriented link diagrams, see Figure 3. The crossing on the left is called a positive Figure 3. A positive crossing and a negative crossing crossing, while the one on the right is called a negative crossing. For a 2-component oriented link diagram D = D1 [ D2, define 1 X lk(D) = "(x); 2 x where the sum is over all the crossings between D1 and D2, and "(x) is the sign of x. Then lk(D) does not changed under oriented Reidemeister moves and define un invariant of 2-components oriented links, known as the linking number. Suppose D is the blackboard diagram of a framed oriented link. Define the writhe of L by X w(L) := "(x): x2C(D) Exercise 1.2. (a) Show that w(L) is an invariant of framed oriented links. (b) Show that the writhe is an invariant of un-oriented knots. (c) Suppose K is an unframed knot, and fr(K) is the set of all framed knots whose underlying unframed knot is K. Show that the map fr(K) ! Z, K0 ! w(K0) is a bijection. Suppose K0 2 fr(K). Let λ be a parallel of K0,which is the result of pushing K off itself along the framed vector field of K0. Show that w(K0) = lk(λ, K). 4 THANG T. Q. LE^ (d) Suppose L = L1 [ L2 be a 2-component oriented link. Define the Gauss map x − y γ : L × L ! S2 = fz 2 R3 j jjzjj = 1g; γ(x; y) = : 1 2 jjx − yjj Show that up to sign, lk(L1;L2) is equal to the degree of γ. The set of all framings of an unframed knot in R3 is naturally identified with Z. For this reason, we also use integers to denote framing a knot. 1.4. Alexander polynomial. Suppose L is an m-component oriented link, and X = S3 nL. A small loop encircling the j-th component is called a meridian of the component, which is defined up to isotopy in the link complement. We choose the orientation of the meridians so that the linking number of the j-th component and its meridian is +1. n From Alexander duality, H1(X; Z) = Z , with generators being the meridians of the links. The map H1(X; Z) ! Z = htj i, mapping each meridian to t, gives rise to a surjective ~ map f : π1(X) ! Z. The corresponding covering X ! X has Z as the group of deck ~ ±1 ±1 transformation. As a result, H1(X; Q) is a Q[Z] ≡ Q[t ]-module. Since Q[t ] is a PID, ~ ±1 and H1(X; Q) is finitely generated over Q[t ] (prove this!), we have Mk ~ ∼ ±1 H1(X; Q) = Q[t ]=(fj); j=1 2 Q ±1 j where each fj [t ], fj fj+1, andQ some of the fj might be 0. The Alexander polynomial 2 Q ±1 k ∆L(t) [t ] of L is defined to be j=1 fk. The Alexander polynomial is defined up to a unit in Q[t±1]. One can choose the unit ±1 normalization such that ∆L(t) 2 Z[t ]. If L is a knot, one can choose a unit normalization of ∆ such that −1 ∆L(t ) = ∆L(t) and ∆L(1) = 1. In particular, for any knot, ∆L(t) =6 0. 1.5. Kauffman bracket. The Kauffman bracket was introduced by [Kau1]. For a good introduction see [Li]. There is a unique function funoriented link diagramsg ! Z[t±1];D ! hDi defined by −1 (1) L = tL+ + t L− (2) L t U = −(t2 + t−2)L; where in the first identity, L; L+;L− are identical except in a ball in which they look like in Figure 4, and in the second identity, the left hand side stands for the union of a link L and the trivial framed knot U in a ball disjoint from L. Here L might be the empty link diagram. In particular, if U is the unknot diagram, the hUi = δ := −(t2 + t−2): COLORED JONES POLYNOMIAL 5 Figure 4. The links L, L+, and L− Lemma 1.3. One has⟨ ⟩ ⟨ ⟩ ⟨ ⟩ (3) −t3 = = −t−3 ⟨ ⟩ ⟨ ⟩ (4) = * + * + (5) = Exercise 1.4. Prove the lemma. Corollary 1.5. There exists a unique invariant ±1=4 ±1=4 foriented framed linksg ! Z[q ];L ! VL 2 Z[q ] such that 1=4 − −1=4 1=2 − −1=2 (6) q VL+ q VL− = (q q )VL0 (7) VLtU = [2]VL 3=4 (8) VL+1 = q VL Here in (6), the links L+;L−;L0 are identical everywhere except for a small balls in which they look like in Figure 5. In (7), L t U is the union of L and a trivial 0-framed knot U which is far away from L. In (8), L+1 is the same as L, with the framing of one of the components increased by +1. Figure 5. From left to right: the links L+;L− and L0 in Equation (6) Here we used the notation [n] for the quantum integer qn=2 − q−n=2 t2n − t−2n [n] := = : q1=2 − q−1=2 t2 − t−2 6 THANG T. Q. LE^ Proof. Suppose L is an oriented framed link with blackboard diagram D. Then #L VL(q) := (−1) hDi t=−q1=4 is an invariant of L, satisfying the requirements of the corollary. ◦ ◦ −(3=4)w(L) If we define V L := q VL, then V is an invariant of oriented unframed links satisfying ◦ ◦ ◦ − 1=2 − −1=2 (9) qV L+ qV L− = (q q )V L0 ◦ ◦ (10) V LtU = [2]V L ◦ Remark 1.6.
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