ABSTRACT. In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum.This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold.

Abstract. Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3-manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n 0, we construct a pair of incommensurable hyperbolic 3-manifolds Nn and N µ n whose volume is approximately n and whose length spectra agree up to length n.Both Nn and N µ n are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that these manifolds have a very thick collar about an essential surface. The two gluing maps differ by a hyper-elliptic involution along this surface. Our proof also involves a new commensurability criterion based on pairs of pants.

The work of Jørgensen and Thurston shows that there is a finite number N (v) of orientable hyperbolic 3-manifolds with any given volume v.In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume v can grow at least factorially fast with v. A similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N (v) in terms of v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.

In this paper, we show that any non-arithmetic hyperbolic 2-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of 3manifolds with non-trivial JSJ-decomposition and rank two fundamental groups. We also show that the only commensurable hyperbolic 2-bridge link complements are the figureeight knot complement and the 6 2 2 link complement. Our work requires a careful analysis of the tilings of R 2 that come from lifting the canonical triangulations of the cusps of hyperbolic 2-bridge link complements. 1 arXiv:1601.01015v2 [math.GT] 11 Jul 2016 Corollary 1.2. Let M be any hyperbolic 2-bridge link complement. If M is non-arithmetic, then M does not irregularly cover any hyperbolic 3-orbifolds (orientable or non-orientable). If M is arithmetic, then M does not irregulary cover any (orientable) hyperbolic 3-manifolds.By combining Corollary 1.2 with the work of Boileau-Weidmann in [3], we get the following characterization of 3-manifolds with non-trivial JSJ-decomposition and rank two fundamental groups. For a more detailed description of this decomposition see Section 5.3.

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