Peripheral Polynomials of Hyperbolic Knots

If K is a hyperbolic knot in the oriented S3, an algebraic component of its character variety containing the holonomy of the complete hyperbolic structure of finite volume of S3 \ K is an algebraic curve (excellent component K). The traces of the peripheral elements of K define polynomial functions in K. These functions are related in pairs by canonical polynomials. These peripheral polynomials, besides producing invariants of the knot, contain geometric information such as volume, Chern-Simons invariant, etc, of the hyperbolic cone manifolds obtained by Dehn-surgery on K. It is shown here that the set of peripheral polynomials of K is determined by just two adjacent peripheral polynomials. It is shown that the curves defined by the peripheral polynomials are all birationally equivalent to K, with only one possible exception. When the knot K is amphicheiral the exception is the canonical peripheral polynomial, which represents a curve two-fold covered by K. The canonical peripheral polynomial relates the trace of the meridian with the trace of the canonical longitude of the knot K, and is essentially a factor of the A-polynomial of K. A practical procedure to compute the polynomial is given.