ScalingStacks

Then the quantity

$$K(D)=(-1)^{x(D)}q^{y(D)-2x(D)}<D>$$

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does not depend on the choice of a diagram $D$ of the oriented link $L$ and is an invariant of $L.$ We denote this invariant by $K(L).$ Up to a simple normalization, $K(L)$ is the Kauffman bracket of link $L$ and equal to the Jones polynomial of $L.$ The Kauffman bracket, $f[L],$ as defined in [Ka], is a Laurent polynomial in an indeterminate $A$ (this $A$ has no relation to the algebra $A$ in Section 2.2 of this paper). One easily sees that setting our $q$ to $-A^{-2}$ and dividing by $(-A^{2}-A^{-2})$ we get $f[L]$:

$$K(L)_{(q=-A^{-2})}=(-A^{2}-A^{-2})f[L]$$

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In this paper we will call $K(L)$ the scaled Kauffman bracket.