# Kaufman Bracket on Links vs Framed Links

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

1. 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.

2. 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 $$\in \mathbb{Z}[A, A^{-1}]$$, and is defined by the recursive formulae