When you are computing the bracket polynomial of the (for example) hopflink, why can you not smooth all the crossings in one go? Why do you have to only first start with one crossing?
For example in this paper http://www.rose-hulman.edu/mathjournal/archives/2006/vol7-n1/paper6/v7n1-6pd.pdf
on page 4, where it is calculating the bracket polynomial of the hopflink, in the second line of the working for A, why doesnt the author just smooth the bottom crossing as well in that step?
You can smooth all the crossings at once. But keep in mind that each crossing is replaced by two smoothings, so the number of smoothed diagrams (disjoint circles with no crossings) doubles with each additional crossing. For example, the Hopf link with $2$ crossings has $2^2 = 4$ smoothings. The standard picture of the trefoil knot with $3$ crossings has $2^3 = 8$ smoothings. (See page 4 of this paper by Dror Bar Natan, but keep in mind that he uses different coefficients for the "0" and "1" smoothings of a crossing.)
Smoothing all the crossings at once is the state-sum approach, and it is very useful. The challenge is to organize the $2^n$ diagrams (for a link with $n$ crossings) in a meaningful way. You have to keep track of how many crossings were smoothed in the "0" vs. "1" way and attach polynomial coefficients accordingly. It's sensible to think of the $0$s and $1$s as coordinates of the vertices of an $n$-dimensional hypercube.
Study the diagram in the linked paper, but try to attach powers of $A$ and $A^{-1}$ (rather than $1$ and $q$) for smoothings and powers of $-A^2 - A^{-2}$ (rather than $q + q^{-1}$) for loops.