Affine cubic surfaces and character varieties of knots

Abstract

It is known that the fundamental group homomorphism π 1 (T 2 ) → π 1 (S 3 \ 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 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 2parameter 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.To Efim Zelmanov on the occasion of his 60th birthday Contents

On the genus two skein algebra

“…Suppose that α q,t is an element in the Arthamonov-Shakirov algebra whose t = q specialisation is equal to α ∈ Sk s (Σ 2,0 ). Then the specialisation ev s 4 ,s 4 (α s 4 ,s 4 • Ψ(0, 0, 0)) is equal to the Jones polynomial of α, viewed as a knot in S 3 under the standard embedding. Proof.…”

The Askey-Wilson algebra and its avatars

“…Kauffman bracket skein module quantizations have been introduced in [37,47] and further studied along our lines of interest for this paper in [39,70,71]. We will now recall some key definitions and results from these investigations.…”

Note on Character Varieties and Cluster Algebras

“…where denotes 8 faces of the octahedral quiver (4.4), {(1,3,5), (2,3,5), (1,4,5), (2,4,5), (1,3,6), (1,4,6), (2,3,6), (2,4,6)}.…”

“…We note that the quantum algebra related to the character varieties of the 4-punctured sphere has a relationship [44] with the C ∨ C 1 double affine Hecke algebra of Cherednik [13], whose polynomial representation gives the Askey-Wilson polynomial (see also [40,48]). As the PSL(2; Z) automorphism of the C ∨ C 1 DAHA was used to construct knot invariant related with the categorification [14] (see also [4]), our cluster algebraic formulation may help for such invariants [33]. It should be noted that the quantum trace map of closed paths could be regarded as the expectation value of a supersymmetric line defect in the N = 2 four dimensional theory, which is conjectured to be written in terms of the quantum Fock-Goncharov coordinates, i.e., the quantum cluster Y -variables [27] (see also [26]).…”