Alternating surgery slopes are algorithmically computable, with explicit computations for many knots and a bound |p/q| ≤ 3g(K)+4 on hyperbolic knots.
Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), 187--194
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The search for alternating surgeries
Alternating surgery slopes are algorithmically computable, with explicit computations for many knots and a bound |p/q| ≤ 3g(K)+4 on hyperbolic knots.