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It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-bridge knots?)

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All two-bridge knots are hyperbolic except for the $(2, k)$ torus links. For an “as simple as possible” (but still fairly difficult) proof, see Theorem 10.17 of Purcell’s book Hyperbolic Knot Theory.

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