Affine cubic surfaces and character varieties of knots

TL;DR

This paper explores the algebraic structure of knot invariants via character varieties, proposing a deformation conjecture linked to cubic surfaces and confirming it for various classes of knots.

Contribution

It connects a deformation conjecture of character varieties to the Brumfiel-Hilden conjecture, confirming the latter for many knot types including torus, 2-bridge, and pretzel knots.

Findings

Confirmed the Brumfiel-Hilden conjecture for infinite classes of knots.

Established the conjecture's stability under connect sums and certain coverings.

Linked the deformation of character varieties to smooth cubic surfaces.

Abstract

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 the complement of a knot $K$ in $S^3$ is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the $\mathrm{SL}_2$-character varieties of the corresponding fundamental groups. In our earlier work [BS16], we proposed a conjecture that the classical restriction map admits a canonical 2-parameter deformation into a smooth cubic surface. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [BH95] 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 certain knot coverings.