Maximal tubes under the deformations of 3-dimensional hyperbolic cone-manifolds

Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-manifold is decreasing and that summed with the cone-angle squared is increasing as we deform the cone-angles. We confirm this near 0 cone-angles for an infinite family of hyperbolic cone-manifolds obtained by Dehn surgeries along the Whitehead link complements. The basic method rests on explicit holonomy computations using the $A$-polynomials and finding the maximal tubes. One of the key tools is the Taylor expansion of a geometric component of the zero set of the $A$-polynomial in terms of the cone-angles. We also show that a sequence of Taylor expansions for Dehn surgered manifolds converges to 1 for the limit hyperbolic manifold.