Links with no exceptional surgeries

Abstract

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.

28 pages, 15 figures. Minor rewording and organizational changes; also added theorem giving a lower bound on the genus of these links

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• A survey of hyperbolic knot theory
• Symmetries and hidden symmetries of $(\epsilon ,d_L)$-twisted knot complements
• Bounds on Pachner moves and systoles of cusped 3-manifolds
• Cusp Volumes of Alternating Knots on Surfaces
• On a family of hyperbolic Brunnian links and their volumes
• Highly twisted diagrams