The A-polynomial 2-tuple of Twisted Whitehead Links

The A-polynomial of a knot in the 3-sphere S 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 S, Zhang defined a polynomial m-tuple link invariant called the Apolynomial 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 S has recently been given in [NZ]. Using this formula, Ni and Zhang [NZ] has computed the A-polynomial of an iterated torus knot explicitly. 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 components 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.