The book "Quantum Invariants: A Study of Knots, 3-Manifolds and Their Sets" by T. Ohtsuki gives the following definitions:
A framed link is the image of an embedding of a disjoint union of annuli into $\mathbb{R}^3$. The underlying link of a framed link is the link obtained by restricting an annulus $S^1\times[0,1]$ to its center line $S^1\times{1/2}$. The framing of a component of a framed link is the isotopy class of framed knots whose underlying knots are equal to the component. The blackboard framing of a diagram on $\mathbb{R}^2$ is the framing parallel to $\mathbb{R}^2$. The following figure goes along with the definitions for clarification:
Then the book states a theorem
Theorem 1.8
Let $L$ and $L'$ be two framed links, and $D$ and $D'$ diagrams of them by blackboard framings. Then, $L$ is isotopic to $L'$ if and only if $D$ is related to $D'$ by a sequence of isotopies of $\mathbb{R}^2$ and the $\mathcal{RI}$, $RII$ and $RIII$ moves in Figure 1.8.
My question is
How does a link being framed lead to a weaker Reidemeister move $\mathcal{RI}$ (and hence, less framed links than links)? In other words, I am asking for a general idea of the proof of the theorem above. The book only gives a "sketch" of the proof, essentially leaving the proof as an exercise.
The Kaufman bracket is not an isotopy invariant for links (unframed), but it is an invariant of framed links. Is there a good explanation why enabling framing reduce the number of links just enough to make the Kaufman bracket an invariant?
To explain my questions, the Reidemeister moves for a link (not framed) are shown in the figure below. We see that the move $RI$ for a link is stronger than the move $\mathcal{RI}$ for a framed link (i.e., $\mathcal{RI}$ is a composition of two $RI$), hence there are more links than there are framed links.
The Kaufman bracket is defined for any link diagram $D$ and takes values in $<D>\in \mathbb{Z}[A, A^{-1}]$, and is defined by the recursive formulae
