Question.
What exactly is known about the image of the Jones-polynomial-function $V$: $\{$ all links $\}$ $\rightarrow$ $\mathbb{Z}[t^{-1/2},t^{1/2}]$?
Are there references explicitly and transparently adressing this question, to the point of desribing the fibers of given Laurent polynomials?
"How" well understood is $\mathrm{im}(V)$?
What are citable references in the literature on this?
Remarks.
(0) Of course, the literature contains a wealth of precise results relevant on this, yet I never saw a reference, let alone a survey addressing this question upfront.
(1) A basic remark to make is that it is known that $0\notin\mathrm{im}(V)$.
(2) I take it that $\mathrm{im}$ can be considered not completely understood, at least in the sense that, notoriously, it is not known whether $V^{-1}(1)$ contains a non-trivial knot.
(3) I in particular need to know as much as conveniently possible about the image of $V$ restricted to the class of all knots$=$(1-component links).