Affine cubic surfaces and character varieties of knots

It is known that the fundamental group homomorphism π1(T2)→π1(S3∖K) induced by the inclusion of the boundary torus into the complement of a knot K in S3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL2-character varieties of the corresponding fundamental groups. In our earlier work [3], we proposed a conjecture that the classical restriction map admits a canonical deformation into a two-parameter family of affine cubic surfaces in C3. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [1] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel-Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel-Hilden conjecture holds is closed under taking connect sums and knot coverings.