How many quadratic fields occur as trace fields of hyperbolic knot complements?

Question

I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs $(F,d)$?

For knot complements, the trace field and invariant trace field are the same, and the trace field is equal to the field generated by the shape parameters of an ideal triangulation. So phrased more topologically, this asks if there are infinitely many knot complements, no two of which share a finite-sheeted cover, where for every shape parameter $s\in\mathbb{C}\setminus\mathbb{R}$, we have $s^2\in\mathbb{R}$.

Update: As @NeilHoffman has pointed out below, the answer to this is not known. So let me ask a weaker question. Are there infinitely many different hyperbolic knot complements with trace fields satisfying the conditions, where we don't mind if only finitely many of the $F$ are realized? This may or may not be any easier, for instance I'm not sure if it's known even if there can be an infinite family sharing the same field.

Update 2: If we relax the original condition so that we allow the manifolds to be link complements rather than just knot complements, the answer is yes. In the following paper by Chesebro and Deblois, that is done with $F=\mathbb{Q}(\sqrt{2})$ and $d=1$: http://www.math.umt.edu/chesebro/AIMCLC.pdf

Answer 1

I think this question is open and even in this narrowly framed context the question still has a number of interesting seemingly weaker questions.

First, as a reference Long and Reid addressed a similar question in section 6 of this paper:

D. D. Long and A. W. Reid, Fields of definition of canonical curves, Interactions between hyperbolic geometry, quantum topology and number theory, Contemporary Math. 541, 247–257, A.M.S. Publications, (2011). (pdf)

As part of there construction of finding ($S^3$) complements with invariant trace fields having class number bigger than 1, they find knot complements that having invariant trace fields which are a quadratic extension of $\mathbb{Q}(\cos(\pi/p))$ ($=\ell_p$ defined above Theorem 1.1 in their paper). Having said that, I don't think it gives a clean, complete answer to your question even for $F=\ell_p$.

Finally, the original question restricted to the case that $F=\mathbb{Q}$ is open. Although A. W. Reid showed the figure eight knot complement is the only knot complement with integral traces and invariant trace field of the form $\mathbb{Q}\sqrt{-d}$ for $d$ a square-free positive integer, the question is wide open for knot complements (in $S^3$) when the integral trace condition is relaxed.