  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 $C\sqrt{n}$ 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
$2^{C\sqrt{n}}.$
