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Commentarii Mathematici Helvetici

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Links with no exceptional surgeries

David Futer (1) and Jessica S. Purcell (2)

(1) Department of Mathematics, Michigan State University, MI 48824, EAST LANSING, UNITED STATES
(2) Department of Mathematics, University of Texas at Austin, TX 78712, AUSTIN, UNITED STATES

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.

Keywords: Dehn surgery, Dehn filling, hyperbolic 3-manifolds, knot complements, link complements