Alternating surgery slopes are algorithmically computable, with explicit computations for many knots and a bound |p/q| ≤ 3g(K)+4 on hyperbolic knots.
Kuperberg, Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization, Pacific J
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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.