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It is known that the fundamental group homomorphism $\pi_1(T^2) \to\pi_1(S^3\setminus K)$ induced by the inclusion of the boundary torus into thecomplement of a knot $K$ in $S^3$ is a complete knot invariant. Many classicalinvariants of knots arise from the natural (restriction) map induced by theabove homomorphism on the $\mathrm{SL}_2$-character varieties of thecorresponding fundamental groups. In our earlier work [BS16], we proposed aconjecture that the classical restriction map admits a canonical 2-parameterdeformation into a smooth cubic surface. In this paper, we show that (modulosome mild technical conditions) our conjecture follows from a known conjectureof Brumfiel and Hilden [BH95] on the algebraic structure of the peripheralsystem of a knot. We then confirm the Brumfiel-Hilden conjecture for aninfinite class of knots, including all torus knots, 2-bridge knots, and certainpretzel knots. We also show the class of knots for which the Brumfiel-Hildenconjecture holds is closed under taking connect sums and certain knotcoverings.

Affine cubic surfaces and character varieties of knots