Summary: Let \((p/q, n)\) denote the orbifold with singular set the two bridge knot or link \(p/q\) and isotropy group cyclic of order \(n\). An algebraic curve \({\mathcal C} [p/q]\) (set of zeroes of a polynomial \(r(x,z)\)) is associated to \(p/q\) parametrizing the representations of \(\pi_1(S^3 - p/q)\) in \(\text{PSL} (2,\mathbb{C})\). The coordinates \(x\), \(z\) are \(\text{trace} (A^2) = x\), \(\text{trace} (AB) = z\) where \(A\) and \(B\) \((\in \text{SL} (2,\mathbb{C})\)) are the images of canonical generators \(a\), \(b\) of \(\pi_1(S^3 - p/q)\). Let \((x_n, z_n)\) be the point of \({\mathcal C} [p/q]\) corresponding to the hyperbolic orbifold \((p/q, n)\). We prove the following result: The (orbifold) fundamental group of \((p/q, n)\) is arithmetic if and only if the field \(Q(x_n, z_n)\) has exactly one complex place and \(\phi (x_n) < \phi (z_n) < 2\) for every real embedding \(\phi : Q(x_n, z_n) \to \mathbb{R}\).
Consider the angle \(\alpha\) for which the cone-manifold \((p/q, \alpha)\) is Euclidean. We prove that \(2 \cos \alpha\) is an algebraic number. Its minimal polynomial (called the \(h\)-polynomial) is then a knot invariant. We indicate how to generalize this \(h\)-polynomial invariant for any hyperbolic knot.
Finally, we compute \(h\)-polynomials and arithmeticity of \((p/q, n)\) with \(p \leq 40\), and \((p/q, n)\) with \(p \leq 99\) \(q^2 \equiv 1 \bmod p\). We finish the paper with some open problems.