We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most 3 hyperbolic knot complements in a cyclic commensurability class. Moreover if two hyperbolic knots have cyclically commensurable complements, then they are fibred with the same genus and are chiral. A characterization of cyclic commensurability classes of complements of periodic knots is also given. In the nonperiodic case, we reduce the characterization of cyclic commensurability classes to a generalization of the Berge conjecture. 57M10, 57M25
We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class. 57M25, 57M10; 57M27
We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such manifolds are finitely covered by great circle link complements.
Abstract. These are notes based on a series of talks that the author gave at the "Interactions between hyperbolic geometry and quantum groups" conference held at Columbia University in June of 2009. Background on hyperbolic manifolds and orbifoldsIn understanding manifolds and their commensurability classes, we will find it extremely helpful to employ orbifolds. A manifold is an object locally modeled on open sets in R n , and an orbifold O is locally modeled on open sets in R n modulo finite groups of Euclidean isometries. That is, each point x ∈ O has a neighborhood modeled onŨ /G, where G is a finite subgroup of SO(n) andŨ is an open ball in R n . A geometric orbifold is the quotient of a simply connected Riemannian manifold X by a discrete subgroup Γ of Isom(X), and we say that O = X/Γ is an X-orbifold. In this case the orbifold fundamental group is the group Γ. There are nongeometric orbifolds, but we will only be concerned with geometric ones here. We will describe the structure of orbifolds through some examples, see [3] for a good description of the details. 1:The "football" is an S 2 -orbifold which is the quotient of the S 2 by the group Z/3Z generated by a rotation of 2π/3 which fixes the north and south poles. The ramification locus of an orbifold O is the set of points where any neighborhood is modeled on an open set in R n modulo a non-trivial group. The ramification locus in this case is two points, which we label 3, since that is the order of the local group. The underlying space |O| of an orbifold O is the space obtained from O by forgetting the orbifold structure, which is S 2 in this case. The two ramification points are both modeled on disks in R 2 modulo a group of rotations, and we call these cone points. The football is commonly denoted as S 2 (3, 3). In general, M 2 (r 1 , ...r n ) is a 2-orbifold with underlying space the 2-manifold M 2 and n cone points of orders r i .
We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic three-torus T 3 . For comparable small volumes, we prove that an area minimizing double bubble in T 3 is the standard double bubble from R 3 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.