ARITHMETIC KNOTS IN CLOSED 3-MANIFOLDS

Abstract

Let Od denote the ring of integers in [Formula: see text]. An orientable finite volume cusped hyperbolic 3-manifold M is called arithmetic if the faithful discrete representation of π1(M) into PSL(2,C) is conjugate to a group commensurable with some Bianchi group PSL (2,Od). If M is a closed orientable 3-manifold, we say a link L ⊂ M is arithmetic if M\L is arithmetic. In the paper, we show that there exist closed orientable 3-manifolds which do not contain anarithmetic knot. Our methods give much more precise informations for non-hyperbolic 3-manifolds. For certain Lens Spaces, we can give fairly complete statements.