Restricting the topology of 1–cusped arithmetic 3–manifolds

Let d be a square-free positive integer, let Od denote the ring of integers in Q. p d/ and let Qd denote the Bianchi orbifold H=PSL.2;Od /. A finite volume, noncompact hyperbolic 3–manifold X is called arithmetic if X and Qd are commensurable, that is to say they share a common finite sheeted cover (see Maclachlan and the second author [15, Chapter 8] for more on this). This paper is concerned with those 1–cusped arithmetic 3–manifolds X DH=€ that actually cover a Bianchi orbifold Qd . If M is a closed orientable 3–manifold, a knot K is called arithmetic, if M nK is arithmetic. Therefore, we are concerned with those arithmetic knots K M for which there is a finite cover M nK!Qd .