Abstract: If Thurston's Geometrization Conjecture is true, then a closed 3-manifold is hyperbolic whenever it satisfies a topological condition, called "hyperbolike". This paper proves a mild diagrammatic condition on a knot or link in S³ under which any non-trivial Dehn filling gives a hyperbolike closed manifold. For a knot K, a non-trivial Dehn filling of K will be hyperbolike whenever a prime, twist-reduced diagram of K has at least 4 twist regions and at least 6 crossings per twist region; the statement for links is similar.

We prove this result using two arguments, one geometric and one combinatorial. The combinatorial argument also proves 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..

Abstract: Let K be a knot that has an unknotting tunnel tau. This paper proves that K admits a strong involution that fixes tau pointwise if and only if K is a two-bridge knot and tau its upper or lower tunnel. One result obtained along the way is a version of the Smith conjecture for handlebodies: the fixed-point set of an orientation-preserving, periodic diffeomorphism of a handlebody is either empty or boundary-parallel.

Abstract: We model interfaces between immiscible fluids as cost-minimizing networks, where "cost" is a weighted length. We consider conjectured necessary and sufficient conditions for when a planar cone is minimizing. In some cases we give a proof; in other cases we provide a counterexample.