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The A-Polynomial 2-Tuple of Twisted Whitehead Links

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THE A-POLYNOMIAL 2-TUPLE OF TWISTED WHITEHEAD LINKS ANH T. TRAN Abstract. We compute the A-polynomial 2-tuple of twisted Whitehead links. As ap- plications, we determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds. 1. Introduction The A-polynomial of a knot in the 3-sphere S3 was introduced by Cooper, Culler, Gillet, Long and Shalen [CCGLS] in the 90’s. It describes the SL2(C)-character variety of the knot complement as viewed from the boundary torus. The A-polynomial carries a lot of information about the topology of the knot. For example, it distinguishes the unknot from other knots [BoZ, DG] and the sides of the Newton polygon of the A-polynomial give rise to incompressible surfaces in the knot complement [CCGLS]. The A-polynomial was generalized to links by Zhang [Zh] about ten years later. For an m-component link in S3, Zhang defined a polynomial m-tuple link invariant called the A- polynomial m-tuple. The A-polynomial 1-tuple of a knot is nothing but the A-polynomial defined in [CCGLS]. The A-polynomial m-tuple also caries important information about the topology of the link. For example, it can be used to construct concrete examples of hyperbolic link manifolds with non-integral traces [Zh]. Finding an explicit formula for the A-polynomial is a challenging problem. So far, the A-polynomial has been computed for a few classes of knots including two-bridge knots C(2n, p) (with 1 ≤ p ≤ 5) in Conway’s notation [HS, Pe, Ma, HL], (−2, 3, 2n + 1)-pretzel knots [TY, GM]. It should be noted that C(2n, p) is the double twist knot J(2n, −p) in the notation of [HS]. Moreover, C(2n, 1) is the torus knot T (2, 2n + 1) and C(2n, 2) is known as a twist knot. A cabling formula for the A-polynomial of a cable knot in S3 has recently been given in [NZ]. Using this formula, Ni and Zhang [NZ] has computed the A-polynomial of an iterated torus knot explicitly. arXiv:1608.01381v1 [math.GT] 3 Aug 2016 In this paper we will compute the A-polynomial 2-tuple for a family of 2-component links called twisted Whitehead links. As applications, we will determine canonical compo- nents of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds. For k ≥ 0, the k-twisted Whitehead link Wk is the 2-component link depicted in Figure 1. Note that W0 is the torus link T (2, 4) and W1 is the Whitehead link. Moreover, Wk is the two-bridge link C(2, k, 2) in Conway’s notation and is b(4k+4, 2k+1) in Schubert’s notation. These links are all hyperbolic except for W0. The A-polynomial 2-tuple of the twisted Whitehead link Wk is a polynomial 2-tuple [A1(M, L), A2(M, L)] given as follows. 2000 Mathematics Subject Classification. Primary 57M27, Secondary 57M25. Key words and phrases. canonical component, cone-manifold, hyperbolic volume, the A-polynomial, twisted Whitehead link, two-bridge link. 1 2 ANH T. TRAN ... Figure 1. The k-twisted Whitehead link Wk. Theorem 1.1. If k =2n − 1 then A1(M, L)=(L − 1)F (M, L) where n n +1+ i n − 1+ i L − 1 2i F (M, L) = − (M − M −1)2i 2i +1 2i +1 L +1 i=0 X n−1 n + i L − 1 2i+1 + (M + M −1)(M − M −1)2i+1 . 2i +1 L +1 i=0 X 2 If k =2n then A1(M, L)=(LM − 1)G(M, L) where n n +1+ i n + i LM 2 − 1 2i+1 G(M, L) = + (M − M −1)2i+1 2i +1 2i +1 LM 2 +1 i=0 X n n + i LM 2 − 1 2i + (M + M −1)(M − M −1)2i . 2i LM 2 +1 i=0 X In both cases we have A1(M, L)= A2(M, L). It should be noted that in the definition of the A-polynomial m-tuple of an m-component link, we discard the irreducible component of the SL2(C)-character variety containing only characters of reducible representations and therefore consider only the nonabelian SL2(C)-character variety of the link group. 3 3 For a link L in S , let EL = S \L denote the link exterior. The link group of L is defined to be the fundamental group π1(EL) of EL. It is known that the link group of the twisted Whitehead link Wk has a standard two-generator presentation of a two-bridge link group π1(EWk )= ha, b | aw = wai, where a, b are meridians depicted in Figure 1 and w is a word in the letters a, b. More precisely, we have w = (bab−1a−1)na(a−1b−1ab)n if k =2n − 1 and w =(bab−1a−1)nbab(a−1b−1ab)n if k =2n. C For a presentation ρ : π1(EWk ) → SL2( ) we let x, y, z denote the traces of the images of a, b, ab respectively. We also let v denote the trace of the image of bab−1a−1. Explicitly, 2 2 2 we have v = x + y + z − xyz − 2. It was shown in [Tr1] that the nonabelian SL2(C)- k character variety of Wk has exactly ⌊ 2 ⌋ irreducible components. For a hyperbolic link, there are distinguished components of the SL2(C)-character variety called the canonical components. They contain important information about the hyperbolic structure of the link. By using the formula of the A-polynomial in Theorem 1.1, we can determine the canonical component of the hyperbolic twisted Whitehead link Wk (with k ≥ 1) as follows. THEA-POLYNOMIALOFTWISTEDWHITEHEADLINKS 3 Theorem 1.2. If k = 2n − 1 then the canonical component of Wk is the zero set of the polynomial (xy − vz)Sn−1(v) − (xy − 2z)Sn−2(v). If k =2n then the canonical component of Wk is the zero set of the polynomial zSn(v)− (xy − z)Sn−1(v). Here the Sk(v) are the Chebychev polynomials of the second kind defined by S0(v) = 1, S1(v)= v and Sk(v)= vSk−1(v) − Sk−2(v) for all integers k. Explicitly, we have k − i S (v)= (−1)i vk−2i k k − 2i 0≤Xi≤k/2 for k ≥ 0, Sk(v)= −S−k−2(v) for k ≤ −2, and S−1(v) = 0. 3 For a hyperbolic link L ⊂ S , let ρhol be a holonomy representation of π1(EL) into P SL2(C). Thurston [Th] showed that ρhol can be deformed into an m-parameter family {ρα1,··· ,αm } of representations to give a corresponding family {EL(α1, ··· ,αm)} of singular complete hyperbolic manifolds, where m is the number of components of L. In this paper we will consider only the case where all of αi’s are equal to a single parameter α. In which case we also denote EL(α1, ··· ,αm) by EL(α). These α’s and EL(α)’s are called the cone-angles and hyperbolic cone-manifolds of L, respectively. We consider the complete hyperbolic structure on a link complement as the cone- manifold structure of cone-angle zero. It is known that for a two-bridge knot or link L 2π there exists an angle αL ∈ [ 3 , π) such that EL(α) is hyperbolic for α ∈ (0,αL), Euclidean for α = αL, and spherical for α ∈ (αL, π) [HLM, Ko1, Po, PW]. Explicit volume formulas for hyperbolic cone-manifolds of knots were known for 41 [HLM, Ko1, Ko2, MR], 52 [Me1], twist knots [HMP] and two-bridge knots C(2n, 3) [HL]. Recently, the volume of double twist knot cone-manifolds has been computed in [Tr3]. We should remark that a formula for the volume of the cone-manifold of C(2n, 4) has just been given in [HLMR1]. However, with an appropriate change of variables, this formula has already obtained in [Tr3], since C(2n, 4) is the double twist knot J(2n, −4). In this paper we are interested in hyperbolic cone-manifolds of links. Explicit volume 2 2 formulas for hyperbolic cone-manifolds of links have been only known for 51 [MV], 62 2 2 [Me2], 63 [DMM] and 73 [HLMR2]. From the proof of Theorem 1.1, we can compute the volume of the hyperbolic cone-manifold of the twisted Whitehead link Wk as follows. Let 2 2 x Sn−1(v) − zSn(v) − (x − z)Sn−2(v) if k =2n − 1 RWk (s, z)= 2 (zSn(v) − (x − z)Sn−1(v) if k =2n, where x = s + s−1 and v =2x2 + z2 − x2z − 2. Theorem 1.3. For α ∈ (0,αWk ) we have π z − (s−2 + 1) Vol E (α)= 2 log dω Wk z − (s2 + 1) Zα where s = eiω/2 and z (with Im z ≥ 0) satisfy R (s, z)=0. Wk Note that the above volume formula for the hyperbolic cone-manifold EWk (α) depends iω/2 on the choice of a root z of RWk (e , z) = 0. In practice, we choose the root z which gives the maximal volume. As a direct consequence of Theorem 1.3, we obtain the following. 4 ANH T. TRAN Corollary 1.4. The hyperbolic volume of the r-fold cyclic covering over the twisted White- head link Wk, with r ≥ 3, is given by the following formula 2π π z − (s−2 + 1) rVol EWk ( )= r 2 log 2 dω r 2π z − (s + 1) Z r where s = eiω/2 and z (with Im z ≥ 0) satisfy R (s, z)=0.
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