Peripheral polynomials of hyperbolic knots

If K is a hyperbolic knot in S 3 , an algebraic component of its character variety containing one holonomy of the complete hyperbolic structure of finite volume of S 3 ∖ K is an algebraic curve K . The traces of the peripheral elements of K define polynomial functions in K , which are related in pairs by polynomials (peripheral polynomials). These are determined by just two adjacent peripheral polynomials. The curves defined by the peripheral polynomials are all birationally equivalent to K , with only one possible exception. The canonical peripheral polynomial relating the trace of the meridian with the trace of the canonical longitude of K, is a factor of the A-polynomial.