Suppose we know the Jones polynomial of some knot, but maybe not specifically which knot. Can any information about the knot be recovered just by knowing its Jones polynomial? Say, for example, the knot's unknotting number, or the minimal number of crossings in projections of the knot.
I am not a knot theorist, so I don't even know where to start. I know the Jones polynomial is most useful as a tool to distinguish knots, but I was curious if anything can be recovered from it.
Depending a bit on what you accept as "information" about the knot, the MathOverflow question linked in the comment answers your question in the positive. In particular, one of the notes in Jones' 1987 article "Hecke Algebra Representations of Braid Groups and Link Polynomials" (on page 368) is that $V_L(1)$ determines the number of components of the link.
That being said, the two pieces of information that you asked for are not determined by the Jones polynomial. Jones remarkes on page 386 of the same paper that the knots $5_1$ and $10_{132}$ share the same Jones polynomial, so the Jones polynomial can not tell you the number of crossings in the knot. Similarly, the unknotting numbers of those two knots are $2$ and $1$ respectively, so that piece of information is also impossible to recover.
