We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field Q(i,\sqrt{2}), but some have integral traces while others do not.Comment: Minor changes following referee's suggestion

Abstract. Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise. A variety of attractive properties follow: such manifolds are virtually fibered; their fundamental groups are LERF; and their geometrically finite subgroups are virtual retracts. Examples of 3-manifolds admitting such a decomposition include augmented link complements. We classify the low-complexity augmented links and describe an infinite family of examples with complements not commensurable to any reflection group in H 3 .

The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields new number theoretic invariants for the character varieties of knot manifolds.Comment: 28 pages, 1 figur

We establish a pair of criteria for proving that most knots obtained by Dehn surgery on a given two-component hyperbolic link lack hidden symmetries. To do this, we use certain rational functions on varieties associated to the link. We apply our criteria to show that among certain infinite families of knot complements, all but finitely many members lack hidden symmetries.A longstanding question in the study of hyperbolic 3-manifolds asks which hyperbolic knot complements, the 3-manifolds obtained by removing a knot from S 3 , have hidden symmetries [37, p. 307]. More recent work of Reid-Walsh [43] and Boileau-Boyer-Cebanu-Walsh [3] relates this to [43, Conjecture 5.2], on commensurability classes of knot complements. We find the original question intriguing simply because hyperbolic 3-manifolds with hidden symmetries are quite common -each manifold that non-normally covers another has them -but hyperbolic knot complements with hidden symmetries seem quite rare. In fact only three are known to have hidden symmetries, a great many are known not to, and no new examples have been found since the publication of [37]. Indeed, the authors Neumann and Reid of [37] later conjectured that no hyperbolic knot complement in S 3 has hidden symmetries, beyond the three already known [16, Problem 3.64(A)].The totality of evidence for this conjecture would still seem to allow for reasonable doubt. Hidden symmetries can be ruled out for (almost) any particular knot complement by straightforward computations using SnapPy [12] and Sage [14] (or Snap, see [11]). For instance, amongst the 300, 000-odd knot complements with at most 15 crossings, only that of the figure-8 has hidden symmetries. Existing tools are harder to apply to families of knot complements and we only know the following classes to lack hidden symmetries: the two-bridge knots other than the figure -8 [43]; the (−2, 3, n)-pretzels [29]; knots obtained from surgery on the Berge manifold [22], [26]; and certain highly twisted pretzel knots with at least five twist regions [34, Prop. 7.5] 1 . Works of Hoffman [25], Boileau-Boyer-Cebanu-Walsh [4], and Millichap-Worden [35] also bear on the question from other directions.Here we provide a plethora of new classes by giving a method for quickly showing that the generic member of certain families of knot complements produced by hyperbolic Dehn surgery lacks hidden symmetries. We were partly inspired to this by [34, Prop. 7.5] and the proof of [22, Theorem 1.1], which are more specific results in the same direction, but we develop a new tool based on the following motto:The cusp parameters of knot complements obtained from a given hyperbolic link complement M by hyperbolic Dehn filling are recorded by rational functions on the character or deformation variety of M that are smooth near the complete structure.

It has been an open question whether all boundary slopes of hyperbolic knots are strongly detected by the character variety. The main result of this paper produces an infinite family of hyperbolic knots each of which has at least one strict boundary slope that is not strongly detected.