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Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in $mathbb S^3$. Then, if $M$ admits an exhaustion by $pi_1$-injective sub-manifolds there exists cantor sets $C_nsubset mathbb S^3$ such that $N_n=mathbb S^3setminus C_n$ is hyperbolic and $N_nrightarrow M$ geometrically.
Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, $Phi (c)$, such that if $K$ is a nontrivial knot i
Given a link in a 3-manifold such that the complement is hyperbolic, we provide two modifications to the link, called the chain move and the switch move, that preserve hyperbolicity of the complement, with only a relatively small number of manifold-l
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (appr
We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_isubset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a
For an arbitrary positive integer $n$ and a pair $(p, q)$ of coprime integers, consider $n$ copies of a torus $(p,q)$ knot placed parallel to each other on the surface of the corresponding auxiliary torus: we call this assembly a torus $n$-link. We c