We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link.
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David Futer, Jessica S. Purcell, Links with no exceptional surgeries. Comment. Math. Helv. 82 (2007), no. 3, pp. 629–664DOI 10.4171/CMH/105