An explicit formula for the \(A\)-polynomial of the knot with Conway's notation \(C(2n,3)\) (Q2827539)

The \(A\)-polynomial of a \(3\)-manifold \(N\) with a single torus boundary is a two-variable polynomial which encodes how eigenvalues of a fixed meridian and longitude are related under representations from \(\pi_1(N)\) into \(\mathrm{SL}_2(\mathbb C)\). This polynomial has found important applications in hyperbolic geometry. However it is relatively difficult to compute and there are very few infinite families of knots for which the \(A\)-polynomial is known.NEWLINENEWLINEIn this paper, the authors give an explicit formula for the \(A\)-polynomial of the knot with Conway's notation \(C(2n,3)\). Their computation relies on an explicit formula of the Riley-Mednykh polynomial associated to the knot.