Canonical components of character varieties of double twist links \(J(2m+1, 2m+1)\) (Q6555314)

Character varieties have been important tools in studying the topology of a complete finite-volume hyperbolic \(3\)-manifold \(M\) with cusps, and canonical components encode a lot of topological information about \(M\).\N\NCharacter varieties of the \(J(k,l)\) double twist knots and links were computed and analyzed by many researchers.\N\NIn this paper, the authors consider the hyperbolic double twist links \(J(2m + 1, 2m + 1)\) which contain the Whitehead link \(J(3, 3)\), and identify their canonical components topologically.\N\NNamely, they prove that the smooth projective model of the canonical component of the \(SL_2 (\mathbb{C})\)-character variety of the double twist link \(J(2m + 1, 2m + 1)\), \(m \geq 1\), is the conic bundle over the projective line \(\mathbb{P}^ 1\) which is isomorphic to the surface obtained from \(\mathbb{P}^ 1\times\mathbb{P}^ 1\) by repeating a one-point blow-up \(6m+3\) times. Equivalently, it is isomorphic to the surface obtained from \(\mathbb{P}^ 2\) by repeating a one-point blow-up \(6m + 4\) times.