Commensurability classes containing three knot complements

COMMENSURABILITY CLASSES OF (−2, 3, n) PRETZEL KNOT COMPLEMENTS

Let K be a hyperbolic (−2, 3, n) pretzel knot and M = S \ K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M . Indeed, if n 6= 7, we show that M is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.

Commensurability Classes of Twist Knots

In this paper we prove that if MK is the complement of a non-fibered twist knot K in S, then MK is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In ad...

Geodesic Surfaces in Knot Complements

Floer Homology and Knot Complements

We use the Ozsváth-Szabó theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the Ozsváth-Szabó Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on ...

Geometric Limits of Knot Complements

We prove that any complete hyperbolic 3–manifold with finitely generated fundamental group, with a single topological end, and which embeds into S is the geometric limit of a sequence of hyperbolic knot complements in S. In particular, we derive the existence of hyperbolic knot complements which contain balls of arbitrarily large radius. We also show that a complete hyperbolic 3–manifold with t...