This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket
skein module is a linear combination of $O(2^g)$ basis elements, with each coefficient a polynomial with at most $n$ nonzero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by $Csqrt{n}$ for some $C.$ From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in $n$ times $2^{Csqrt{n}}.$
